This document contains the step-by-step work to solve the equation 9x + 2y = 18 for y. It begins with the original equation, subtracts 9x from both sides, and then divides both sides by 2 to isolate y, resulting in the solution y = -9x/2 + 9.
This document provides instructions for factoring a trinomial. It gives the example of factoring x^2 + 7x - 18. The steps shown are writing out the trinomial, finding the greatest common factor of the first and last terms, then using that to write the trinomial as the product of two binomials, (x - 2)(x + 9), which provides the factored form of the original trinomial.
This document shows the step-by-step algebraic manipulation of the equation 9x + 7y - 1 = 0. Through adding 1 to both sides and subtracting 9x, the equation is transformed into y = -9x/7 + 1/7, representing the line that satisfies the original equation.
This document provides steps to factor the binomial x^2+3x+18 into two binomials. It first writes out the binomial to be factored. It then uses the grouping method, factoring the binomial into (x+3)(x-6).
The document provides steps for factoring a trinomial expression, x^2+7x-18, by using two binomials. It shows setting up the trinomial with negative signs underneath and canceling out pairs of terms to reveal the factors (x+9) and (x-2), which when multiplied give the original trinomial x^2+7x-18.
The document contains mathematical equations for solving a quadratic equation. It begins with the standard form of a quadratic equation, then isolates the x term on one side and combines like terms on the other side to solve for x, resulting in x=3.
This document provides instructions for factoring the trinomial X^2 + 7x - 18. It shows that this trinomial can be factored into the form (x - 9)(x + 2), with the first term (x - 9) representing the zero of the parabola where x = 9, and the second term (x + 2) representing the other zero where x = -2. The steps for factoring the trinomial into this form are not shown.
This document provides examples of subtracting polynomials. It begins with examples of subtracting terms like x+7 and 3x+9. It then shows how to subtract polynomials by removing corresponding terms and using the opposite sign. For example, (3x^2 + x + 2) - (2x^2 - x + 3) is solved by removing the 2x^2 terms and changing the sign of the remaining terms to get x^2 + x - 1. It provides several more examples and problems for students to solve, including finding the area of a shaded region between a rectangle and square using polynomial subtraction.
This short document contains two mathematical expressions: 2x1, representing two times one, and 3x2, representing three times two. It presents two simple multiplication problems without showing the solutions.
This document provides instructions for factoring a trinomial. It gives the example of factoring x^2 + 7x - 18. The steps shown are writing out the trinomial, finding the greatest common factor of the first and last terms, then using that to write the trinomial as the product of two binomials, (x - 2)(x + 9), which provides the factored form of the original trinomial.
This document shows the step-by-step algebraic manipulation of the equation 9x + 7y - 1 = 0. Through adding 1 to both sides and subtracting 9x, the equation is transformed into y = -9x/7 + 1/7, representing the line that satisfies the original equation.
This document provides steps to factor the binomial x^2+3x+18 into two binomials. It first writes out the binomial to be factored. It then uses the grouping method, factoring the binomial into (x+3)(x-6).
The document provides steps for factoring a trinomial expression, x^2+7x-18, by using two binomials. It shows setting up the trinomial with negative signs underneath and canceling out pairs of terms to reveal the factors (x+9) and (x-2), which when multiplied give the original trinomial x^2+7x-18.
The document contains mathematical equations for solving a quadratic equation. It begins with the standard form of a quadratic equation, then isolates the x term on one side and combines like terms on the other side to solve for x, resulting in x=3.
This document provides instructions for factoring the trinomial X^2 + 7x - 18. It shows that this trinomial can be factored into the form (x - 9)(x + 2), with the first term (x - 9) representing the zero of the parabola where x = 9, and the second term (x + 2) representing the other zero where x = -2. The steps for factoring the trinomial into this form are not shown.
This document provides examples of subtracting polynomials. It begins with examples of subtracting terms like x+7 and 3x+9. It then shows how to subtract polynomials by removing corresponding terms and using the opposite sign. For example, (3x^2 + x + 2) - (2x^2 - x + 3) is solved by removing the 2x^2 terms and changing the sign of the remaining terms to get x^2 + x - 1. It provides several more examples and problems for students to solve, including finding the area of a shaded region between a rectangle and square using polynomial subtraction.
This short document contains two mathematical expressions: 2x1, representing two times one, and 3x2, representing three times two. It presents two simple multiplication problems without showing the solutions.
This document contains 12 algebra word problems involving systems of linear equations. Students are asked to solve for the variables x and y. The problems cover a variety of equation types including single variable equations, addition/subtraction of terms, and multiplication of terms by constants. Solving the systems of equations requires combining like terms and eliminating variables.
This document provides steps to factor the trinomial 6x^2 + 3x - 3 into binomials. It uses a factoring method of finding two binomial factors whose product is the original trinomial. It determines the factors are (x+1) and (6x-3), so the factored form is (x+1)(6x-3).
This document contains 8 systems of linear equations with 2 unknown variables (x and y) each. Solving these simultaneous equations would yield the unique values of x and y that satisfy all 8 systems simultaneously, if a solution exists.
The document shows simple addition and subtraction equations with single-digit numbers, with the answers being the results of adding or subtracting one number from another. A few examples are 2 + 1 = 3, 2 - 1 = 1, and 3 - 0 = 3, demonstrating basic math operations with small whole numbers.
1. The document proves through mathematical induction that the formula nj=1 j^2 = n(n+1)(2n+1)/6 for the sum of squares from 1 to n is true for all positive integers n.
2. It also proves that a 2n x 2n chessboard with one square missing can be covered with L-shaped pieces that each cover three squares, again using mathematical induction.
3. Mathematical induction is used to prove both formulas by showing the base case holds and assuming the formula is true for an integer n to prove it is true for n+1.
The document shows the step-by-step solution to the equation 2(x + 7) = 13. It distributes the 2 to get 2x + 14 = 13, then subtracts 14 from both sides to get 2x = -1, and finally divides both sides by 2 to find the solution x = -1/2.
The document is a math report that contains 14 quadratic equations solved by factoring. Each equation is presented with the factored form set equal to zero to solve for the roots. The roots provided for each equation are the solutions to the quadratic equation. The report ends with the names of the 9 group members who worked on the math problems.
Factorización aplicando Ruffini o Método de EvaluaciónWuendy Garcia
(1) The document provides two examples of factorizing polynomial expressions.
(2) The first polynomial, x3 + 5x2 - 2x - 24, is factorized into (x-2)(x+3)(x+4).
(3) The second polynomial, 2x3 - 3x2 - 11x + 6, is factorized into (2x - 1)(x + 2)(x - 3).
This document shows the step-by-step working of distributing a negative sign when multiplying a number by a binomial expression. It starts with the expression -3(6x + 1) and through distributing the negative sign, arrives at the equivalent expression -18x - 3 in 3 lines.
This document lists multiplication facts for the number 2. It shows that 2 multiplied by single digit numbers 1 through 12 equals the products 2 through 24 in order.
The document shows the step-by-step work of solving the equation 7x - 9y + 14 = 0 for y in terms of x. Through adding and subtracting like terms, the equation is isolated to 7y = 9x - 14, which when divided by 7 results in the solution y = 9x/7 - 2.
This document provides examples and instructions for subtracting polynomials. It begins with examples of subtracting terms like x+7 and 3x+9. It then shows how to subtract polynomials by removing common terms and combining like terms, such as (3x^2 + x + 2) - (2x^2 - x + 3). The document ends with homework assignments involving subtracting various polynomials.
This document contains color assignments for various animals and people completed by Katie Brown for her Period 1 DPI2 class. It lists the names of a sleeping cat, seal, grizzly, parrot, herself twice, a bear, tigré, turtle, and tigré dos along with the instruction "Color" for each entry.
This document outlines steps to improve energy efficiency in a home. It recommends replacing incandescent light bulbs with compact fluorescent lights (CFLs) to reduce energy usage. It also suggests adjusting the thermostat setting to use heating and cooling more efficiently when home. Additional recommendations include installing programmable thermostats, improving insulation, and sealing air leaks.
El documento presenta una serie de ecuaciones lineales de la forma y=mx+b, donde m representa la pendiente y b el corte con el eje y. Se muestran 12 pares de ecuaciones con diferentes pendientes y cortes con el eje y.
The document presents several pairs of lines with the same slope (m) but different y-intercepts (b), demonstrating that parallel lines have the same slope but different y-intercepts. It includes 10 pairs of lines in standard y=mx+b form to illustrate that while the m term is the same between parallel lines, the b term can vary between each pair without affecting their parallel nature.
In our March 2015 presentation on the state of the Australian and global economy, we discuss the key themes of the year ahead. Specifically, we look at the prospect of rising interest rates in the United States, the impact of quantitative easing in Europe and the ongoing economic slowdown in China. We also consider the state of the Australian economy and how financial markets are positioned.
This document contains the equations of 6 sets of lines. Each set contains 2 lines with their respective slope-intercept forms. The slopes and y-intercepts of the lines in each set are provided. The document concludes that if the lines in any set are intersecting, non-parallel lines, then the product of their slopes will not equal -1.
The document provides 3 examples of combining like terms in algebraic expressions. Each example shows identifying like terms, combining their coefficients, and obtaining a final simplified expression. The examples involve adding and combining terms with variables x, a, and b.
The document discusses resilience and how to build and maintain resilience. It defines resilience as the ability to bounce back from emotional trauma and adversity. Resilience can be developed and is a renewable resource. The document suggests that understanding your baseline well-being, triggers, and responses can help build resilience. Specific strategies mentioned to refill your resilience "bucket" include conscious breathing, maintaining perspective, connecting with others, focusing on flow activities, practicing gratitude, and finding purpose and meaning.
This document contains 12 algebra word problems involving systems of linear equations. Students are asked to solve for the variables x and y. The problems cover a variety of equation types including single variable equations, addition/subtraction of terms, and multiplication of terms by constants. Solving the systems of equations requires combining like terms and eliminating variables.
This document provides steps to factor the trinomial 6x^2 + 3x - 3 into binomials. It uses a factoring method of finding two binomial factors whose product is the original trinomial. It determines the factors are (x+1) and (6x-3), so the factored form is (x+1)(6x-3).
This document contains 8 systems of linear equations with 2 unknown variables (x and y) each. Solving these simultaneous equations would yield the unique values of x and y that satisfy all 8 systems simultaneously, if a solution exists.
The document shows simple addition and subtraction equations with single-digit numbers, with the answers being the results of adding or subtracting one number from another. A few examples are 2 + 1 = 3, 2 - 1 = 1, and 3 - 0 = 3, demonstrating basic math operations with small whole numbers.
1. The document proves through mathematical induction that the formula nj=1 j^2 = n(n+1)(2n+1)/6 for the sum of squares from 1 to n is true for all positive integers n.
2. It also proves that a 2n x 2n chessboard with one square missing can be covered with L-shaped pieces that each cover three squares, again using mathematical induction.
3. Mathematical induction is used to prove both formulas by showing the base case holds and assuming the formula is true for an integer n to prove it is true for n+1.
The document shows the step-by-step solution to the equation 2(x + 7) = 13. It distributes the 2 to get 2x + 14 = 13, then subtracts 14 from both sides to get 2x = -1, and finally divides both sides by 2 to find the solution x = -1/2.
The document is a math report that contains 14 quadratic equations solved by factoring. Each equation is presented with the factored form set equal to zero to solve for the roots. The roots provided for each equation are the solutions to the quadratic equation. The report ends with the names of the 9 group members who worked on the math problems.
Factorización aplicando Ruffini o Método de EvaluaciónWuendy Garcia
(1) The document provides two examples of factorizing polynomial expressions.
(2) The first polynomial, x3 + 5x2 - 2x - 24, is factorized into (x-2)(x+3)(x+4).
(3) The second polynomial, 2x3 - 3x2 - 11x + 6, is factorized into (2x - 1)(x + 2)(x - 3).
This document shows the step-by-step working of distributing a negative sign when multiplying a number by a binomial expression. It starts with the expression -3(6x + 1) and through distributing the negative sign, arrives at the equivalent expression -18x - 3 in 3 lines.
This document lists multiplication facts for the number 2. It shows that 2 multiplied by single digit numbers 1 through 12 equals the products 2 through 24 in order.
The document shows the step-by-step work of solving the equation 7x - 9y + 14 = 0 for y in terms of x. Through adding and subtracting like terms, the equation is isolated to 7y = 9x - 14, which when divided by 7 results in the solution y = 9x/7 - 2.
This document provides examples and instructions for subtracting polynomials. It begins with examples of subtracting terms like x+7 and 3x+9. It then shows how to subtract polynomials by removing common terms and combining like terms, such as (3x^2 + x + 2) - (2x^2 - x + 3). The document ends with homework assignments involving subtracting various polynomials.
This document contains color assignments for various animals and people completed by Katie Brown for her Period 1 DPI2 class. It lists the names of a sleeping cat, seal, grizzly, parrot, herself twice, a bear, tigré, turtle, and tigré dos along with the instruction "Color" for each entry.
This document outlines steps to improve energy efficiency in a home. It recommends replacing incandescent light bulbs with compact fluorescent lights (CFLs) to reduce energy usage. It also suggests adjusting the thermostat setting to use heating and cooling more efficiently when home. Additional recommendations include installing programmable thermostats, improving insulation, and sealing air leaks.
El documento presenta una serie de ecuaciones lineales de la forma y=mx+b, donde m representa la pendiente y b el corte con el eje y. Se muestran 12 pares de ecuaciones con diferentes pendientes y cortes con el eje y.
The document presents several pairs of lines with the same slope (m) but different y-intercepts (b), demonstrating that parallel lines have the same slope but different y-intercepts. It includes 10 pairs of lines in standard y=mx+b form to illustrate that while the m term is the same between parallel lines, the b term can vary between each pair without affecting their parallel nature.
In our March 2015 presentation on the state of the Australian and global economy, we discuss the key themes of the year ahead. Specifically, we look at the prospect of rising interest rates in the United States, the impact of quantitative easing in Europe and the ongoing economic slowdown in China. We also consider the state of the Australian economy and how financial markets are positioned.
This document contains the equations of 6 sets of lines. Each set contains 2 lines with their respective slope-intercept forms. The slopes and y-intercepts of the lines in each set are provided. The document concludes that if the lines in any set are intersecting, non-parallel lines, then the product of their slopes will not equal -1.
The document provides 3 examples of combining like terms in algebraic expressions. Each example shows identifying like terms, combining their coefficients, and obtaining a final simplified expression. The examples involve adding and combining terms with variables x, a, and b.
The document discusses resilience and how to build and maintain resilience. It defines resilience as the ability to bounce back from emotional trauma and adversity. Resilience can be developed and is a renewable resource. The document suggests that understanding your baseline well-being, triggers, and responses can help build resilience. Specific strategies mentioned to refill your resilience "bucket" include conscious breathing, maintaining perspective, connecting with others, focusing on flow activities, practicing gratitude, and finding purpose and meaning.
This document contains titles of artworks and the artist's name, Katie Brown. It lists 6 pieces including "S-Curves & Smoke", "Burning Rose", "Alana", "Flowers", and "Kathy". Each entry also notes that the artwork was created in color and black and white.
The document discusses linear equations and their slopes. It shows that the slopes of two perpendicular lines will always have a product of -1. Specifically, it provides examples of linear equations and calculates the slopes and their products, demonstrating that the product is consistently -1, proving that perpendicular lines have slopes with a product of -1.
While some important issues women’s wellness and health are paramount considerations for many, coverage varies widely because they’re mistakenly overlooked. Visit : http://www.midfloridabcbs.com
In July 2013 we held one of our regular client functions where we discussed the state of the Australian and global economies and our outlook for the next year. If you were unable to attend the function you can view the presentation below.
This document provides step-by-step work to factor the expression 1x^2 - 9. It begins with the expression, multiplies -9 to both sides, then factors -9 into 3*3. This leads to factoring the entire expression as (x-3)(x+3), showing the two factors that multiply to the original expression.
This document contains titles and attribution for 9 color photographs taken by Katie Brown for her Period 1 class. The photographs depict various nature subjects including lilacs, a petal, an unidentified fluff thing, two stumps, and several images of flowers. Katie Brown took all photos and they were works completed for a class assignment.
El documento contiene 12 conjuntos de ecuaciones lineales. Cada conjunto contiene dos ecuaciones lineales de la forma y=mx+b que representan rectas. Las ecuaciones varían en sus pendientes (m) y ordenadas al origen (b).
This artist chose to share a drawing they created and photographed, explaining that they drew the picture using pencils and an eraser, took a photo of it, and then edited the photo in Photoshop by increasing the contrast and removing the background. The artist notes they would adjust some elements like the editing and shading if redoing the piece.
El documento contiene 10 ecuaciones lineales de la forma y=mx+b, donde se muestran diferentes pendientes m y ordenadas al origen b, incluyendo fracciones como coeficientes de la pendiente.
The document discusses energy efficiency upgrades for a home, including replacing incandescent light bulbs with compact fluorescent light bulbs (CFL), installing a programmable thermostat, adding insulation in the attic, sealing air leaks, and upgrading to more efficient appliances. The homeowner would save money on utility bills through lower energy usage after implementing the recommended upgrades.
The document shows the steps to solve the equation 3x + 6y - 8 = 4 for x. It begins with adding 8 to both sides, then subtracting 6y from both sides. This leaves 3x = -6y + 12, which is then divided by 3 to isolate x as x = -2y + 4.
The rules for solving inequalities are the same as for equations, except when dividing by a negative number, the inequality sign flips. This is demonstrated through examples of solving various types of inequalities, including those with fractions, variables on both sides, and using the distributive property. The key point is checking if the number divided by is positive or negative to determine if the inequality sign remains the same or flips.
The document provides examples of solving linear equations in three steps: 1) combining like terms, 2) using the inverse operation to isolate the variable, and 3) dividing to solve for the variable. In example 1, the equation 2x + 6x = -24 is solved to get x = -3. In example 2, the equation 8a + 3 - 2a = -17 is solved to get a = -10/3.
The document shows the step-by-step work of solving the equation 3 – 5(x + 1) = 21. It distributes the -5, combines like terms, and isolates x to find that the solution is x = 23/5. The key steps are to always distribute first, distribute the negative sign correctly, and show the work clearly at each step of the solution.
The document shows the step-by-step work of solving the equation -2(4x + 5) +3 = -8. It begins with distributing the -2, combining like terms, and performing inverse operations until arriving at the solution of x = 1/8.
This document shows the step-by-step work of solving the equation (3x - 6) = 24 for x. It begins with distributing the -1, then combining like terms and solving for x by first adding 6 to both sides and then dividing both sides by -3, resulting in the solution of x = 6.
The document shows the step-by-step working of the expression 4(2x - 7), which equals 8x - 28 when fully simplified. It cautions that a common mistake is forgetting that the minus sign belongs to the 7 term, and not treating 2x - 7 as a single term.
This document provides an example of distributing a term over a parenthesis in an algebraic expression. It shows the steps of distributing the coefficient 7 over the terms in the parenthesis (x + 6), resulting in the equivalent expression of 7x + 42.
The document provides 7 examples of solving linear equations by performing inverse operations to isolate the variable. Each example shows the step-by-step work including adding, subtracting, multiplying, or dividing both sides of the equation by the same number to simplify it until the variable is alone on one side of the equation. The examples demonstrate solving equations for various types of linear expressions involving addition, subtraction, multiplication, and division of the variable.
This document describes factoring the quadratic expression 1x^2 - 12x + 32. It shows the steps of multiplying the expression by 1, combining like terms, identifying the factors of the constant term 32 as 1 x 32 and 2 x 16, and determining that the factors of the expression are (x - 4)(x - 8) which results in the fully factored form.
The document shows the step-by-step factorization of the polynomial 3x^2 - 8x - 3. It factors the expression into (3x + 1)(x - 3) by first finding the greatest common factor of -9, then determining the signs of the factors based on the leading coefficient, and finally dividing both factors by the leading coefficient of 3 to complete the factorization.
This document discusses factorizing the quadratic expression 1x^2 - 2x - 24. It shows the steps of multiplying the expression by 1, finding the factors of -24, and determining that the factors that combine to give -24 are (x-6)(x+4), following the sign of the larger number. The expression is therefore factorized as (x-6)(x+4).
This document describes factorizing the quadratic expression x^2 + 1x - 20. It shows the steps of multiplying, finding the factors of -20, and determining that the expression can be fully factorized as (x - 4)(x + 5).
This document shows the step-by-step factorization of the expression 1x^2 + 6x + 8. It begins with the original expression and shows multiplying and adding like terms. The expression is then factored into (x + 2)(x + 4), showing the work and reasoning for combining the factors.
Two lines are perpendicular if their slopes have a product of -1. The slopes of perpendicular lines will always satisfy the property of having a product of -1.
El documento contiene varias ecuaciones de líneas. Cada sección presenta dos ecuaciones de líneas, una en función de y = mx + b y la otra en función de y = bx + m. Las ecuaciones describen líneas con diferentes pendientes y ordenadas al origen.
El documento contiene varias ecuaciones lineales de la forma y=mx+b con diferentes pendientes m y ordenadas al origen b. Cada par de ecuaciones tiene la misma pendiente m pero diferente ordenada al origen b, indicando líneas paralelas.
This document contains 10 linear equations in the form of y=mx+b. The equations represent lines with different slopes and y-intercepts, except for the last two lines which are coincident since they have the same slope of -4 and y-intercept of 0.
Este documento presenta varias ecuaciones lineales de la forma y = mx + b, incluyendo y = 3x + 4, y = 5x + 1, y = 2/3x + 6, y = -3/2x - 8, y = 2x + 6 y y = 7x + 3.