A17-2and7-3 worksheet
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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.
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The property illustrated by the expressions 8 + [6 + (-7)] = (8+6) + (-7) and 12 + 6 = 18 is the associative property of addition.
If 4r + 20 (7h) =36, and r=5h, then 20h + 20 (7h) =36 illustrates substituting 5h for r in the original equation.
If x+4 = 15, then x+4 – 4 = 15 – 4 illustrates subtracting 4 from both sides of the original equation to isolate the variable x.
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1) The document solves two systems of linear equations.
2) For the first system, it finds that the solution is x=8-2y and y=3.
3) For the second system, it also finds the solution is y=3.
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1. The document presents formulas for expanding and factorizing binomial expressions, including squaring binomials, multiplying binomials, and cubing binomials.
2. Formulas are given for expanding (a ± b)2, (a ± b)(a ∓ b), and (a + b)(a + c) as well as cubing expressions of the form (a ± b)3.
3. Examples of applying the formulas are shown, such as expanding (5x + 1)2 and factorizing (3x - 5)3.
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This document contains a math problem involving complex numbers and De Moivre's formula. It asks for the value of a complex number written as [2(cos(π/10) + i sin(π/10))]5. The solution uses De Moivre's formula to rewrite the expression in exponential form, and then evaluates it to get the final answer of 32i.
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1. The document contains a math exercise with 26 quadratic equations.
2. Students are asked to solve each quadratic equation for the set of solutions (S).
3. The equations cover a range of standard quadratic forms including factored, unfactored, and equations set to other expressions.
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This document provides an overview and activities on solving quadratic equations by factoring. It begins by defining quadratic equations and their standard form. Several activities are presented to practice identifying quadratic equations, rewriting them in standard form, and factoring trinomials of the form x^2 + bx + c. The final activity involves factoring quadratic equations to determine their roots. The document aims to build mastery of skills needed to solve quadratic equations using factoring techniques.
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2. Tables are included showing the first ten perfect squares and cubes.
3. Questions at the end provide practice evaluating additional expressions and problems involving squares, square roots, cubes and cube roots.
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2. To square a binomial, square each term and multiply the unlike terms by 2.
3. Examples are provided of squaring binomials like (x + 6)2 = x2 + 12x + 36 and factoring trinomials into perfect square forms like (x - 2)2.
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1. The document is a math guide with exercises on quadratic equations for 9th grade students.
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2) Steps for completing the square to put a quadratic equation into standard form for a hyperbola, including an example starting with 4y^2 - 9x^2 - 18x - 16y = 29.
3) An example of graphing a parabola given by the equation x = -y^2 + 2y + 15, identifying
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1) It shows how to write the quadratic equation x^2 - 8x + 12 = 0 given the roots are 2 and 6.
2) It shows how to write the quadratic equation 2x^2 - 9x + 4 = 0 given the roots are 4 and 1/2.
3) The key steps are finding the sum and product of the roots, and using those values to write the quadratic equation in the form x^2 - (sum of roots)x + (product of roots) = 0.
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This document contains examples of algebraic expressions and equations. It shows how to combine like terms, factor expressions, and solve simple equations. For example, it shows that (x + 2)(x + 3) can be factored as x^2 + 5x + 6. It also provides step-by-step workings for subtracting expressions like 2y - (-5y) = 2y + 5y = 7y.
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Here are the steps to solve this equation by factorising:
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1. The solution to a system of equations is a point, while the solution to a system of inequalities is a shaded region on a graph.
2. Parallel lines have the same slope and never intersect, so they have no solution. Coinciding lines are the same single line, so they have infinitely many solutions.
3. The document provides examples of solving systems of equations and inequalities through graphing, substitution, and elimination. The solutions are given as points or descriptions of regions.
PC Test 2 study guide 2011
1. The document is a study guide for a precalculus test that covers topics involving vectors, complex numbers, and their representations and operations.
2. It lists 15 problems involving finding magnitudes and direction of vectors, vector components, dot and cross products, complex numbers in rectangular and polar form, and converting between polar and rectangular coordinates.
3. The problems cover basic vector and complex number calculations, representations, properties and conversions.
A2 Test 2 study guide with answers (revised)
The document is an algebra 2 study guide that provides practice problems for graphing and solving various types of inequalities and systems of equations, including:
1) Graphing linear and compound inequalities on number lines and coordinate planes
2) Solving absolute value equations and inequalities
3) Solving systems of linear equations through graphing, substitution, and elimination
The study guide contains over 50 problems to help students prepare for an algebra 2 test.
A2 Test 2 study guide with answers
This document provides a study guide with answers for an Algebra 2 Test 2. It includes instructions to graph and solve various types of inequalities, systems of equations and absolute value equations, including linear and compound inequalities, systems of linear equations solved by graphing, substitution and elimination, and absolute value equations and inequalities.
A1 Test 2 study guide with answers 2011
The document is an algebra 1 test study guide containing 30 problems involving graphing and solving various types of inequalities, including linear, compound, absolute value, and graphing solutions on number lines. Key topics covered include writing inequalities from graphs, solving single-variable and compound inequalities, graphing linear inequalities on a coordinate plane, and solving absolute value equations and inequalities.
A1 Test 2 study guide
1. The document provides a study guide for an Algebra 1 Test 2 with questions about: graphing and writing inequalities on number lines, solving linear inequalities and compound inequalities, graphing linear inequalities on the xy-plane, and solving absolute value equations and inequalities.
2. Questions involve skills like graphing inequalities, writing inequalities from graphs, solving one-step and two-step inequalities, and solving absolute value equations and inequalities.
3. The study guide covers key Algebra 1 concepts to help prepare for Test 2.
PCExam 1 practice with answers
1. This practice exam covers topics like complex numbers, functions, limits, and graphing.
2. It asks students to choose problems involving adding, multiplying, composing, and finding inverses of various functions like f(x)=9x^2+1 and g(x)=x-1.
3. Students also must graph and classify functions, evaluate limits, and perform operations on complex numbers, plotting them on a plane. The exam covers concepts in precalculus.
PCExam 1 study guide answers
1. The document provides examples and definitions for various mathematical concepts including natural numbers, integers, rational numbers, real numbers, imaginary numbers, and complex numbers.
2. It also includes problems involving composition of functions, finding inverses of functions, operations on complex numbers, graphing and classifying functions, and evaluating limits.
3. Examples cover topics like composition and inverses of functions, operations on complex numbers, classifying functions as linear, quadratic, constant, etc. and their domains and ranges, and evaluating limits including ones that are undefined.
PC Exam 1 study guide
This study guide covers topics in precalculus including:
1) Examples of natural numbers, integers, rational numbers, and real numbers.
2) Composition of functions including finding f(g(x)) and g(f(x)) for given functions f(x) and g(x).
3) Finding inverses of functions and verifying inverses by composition.
4) Operations on complex numbers including plotting, finding modulus, distance, midpoint, addition, subtraction, multiplication, and division.
5) Graphing functions, classifying function types, and stating domains and ranges.
6) Finding limits of functions as x approaches values.
PC 1 continuity notes
A continuous function is defined as one that is defined at a point c, where the limit exists and the function approaches the same y-value from both sides of point c. A function may be discontinuous due to an infinite jump, a jump, or a point discontinuity. Rational functions are defined as the quotient of two polynomial functions. They have asymptotes which occur where the denominator is zero, creating either a vertical or horizontal asymptote in the graph. The end behavior of functions can be determined by considering whether they are increasing or decreasing over different intervals.
PC 1 continuity
This document discusses continuity and discontinuities in functions, defining continuous functions as smooth curves where the domain includes all real numbers. It presents tests for continuity, requiring a function to be defined at a point, have an existing function value, and approach the same y-value from both sides. Critical points and extrema are defined as maximums where a function changes from increasing to decreasing, and minimums where it changes from decreasing to increasing. Rational functions are described as the quotient of two polynomial functions, with limited domains and possible vertical asymptotes.
A1 3 linear fxns
The document discusses the slope-intercept form of a linear equation, y=mx+b. It explains that x represents the input, y represents the output, m is the slope of the line, and b is the y-intercept. It provides instructions for graphing a linear equation in slope-intercept form, which is to first plot the y-intercept and then use the slope to rise and run to the next point, connecting the points with a line.
A1 3 linear fxns notes
The document discusses slope and linear equations. It defines slope as rise over run and provides examples of calculating slope from graphs and ordered pairs. It also defines the standard form of a linear equation as y=mx+b, where m is the slope and b is the y-intercept. The document explains how to graph lines from their equations in standard form by plotting the y-intercept and using the slope to rise and run to the next point.
A1 2 linear fxns
This document discusses the slope-intercept form of a linear equation, y=mx+b. It explains that x represents the input, y represents the output, m is the slope of the line, and b is the y-intercept. It provides instructions for graphing a linear equation in slope-intercept form, which is to first plot the y-intercept, then use the slope to rise and run to the next point and connect them with a line.
A1 2 linear fxns notes
This document discusses intercepts and slope of lines in algebra 1. It defines x-intercepts as where the line crosses the x-axis and y-intercepts as where it crosses the y-axis. It provides the equations for finding each, and includes two examples of finding intercepts and graphing lines. It also defines slope as rise over run and asks what the slope is of a given graph, with homework assigned on slope of lines and graphing lines intuitively.
A2 2 linear fxns notes
This document provides instructions and examples for finding the slope and y-intercept of a line from its equation, ordered pairs, or graph and using them to graph the line. It explains that slope is defined as rise over run and is used along with the y-intercept to graph lines by standard form, plotting the y-intercept and using slope to find successive points. Exercises are provided to have students practice finding slope, y-intercept, and graphing lines from different representations.
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A17-2and7-3 worksheet
- 1. Algebra Problems. code=1296710858 2/2/11 10:03 PM Name:__________________________ Algebra Practice Problems Date:__________________________ Worksheet generated at www.math.com 1.) - x - 4y = - 36 2.) - 2x - 2y = - 18 - 4x + 4y = 36 5x + 2y = 18 3.) 5x + 4y = - 16 4.) 5x - 5y = - 45 - 2x + 4y = 12 - 5x - 5y = 45 5.) - 4x + 4y = 36 6.) x + y = 13 3x + 4y = 15 x + 3y = 25 7.) - x - 4y = 49 8.) 3x + 4y = - 45 - 3x - 16y = 187 x + 16y = - 147 9.) - 3x + y = - 21 10.) - 2x + y = - 12 x + 4y = 46 5x + y = 58 http://www.math.com/cgi-bin/mathf_sp.pl Page 1 of 2
- 2. Algebra Problems. code=1296710858 2/2/11 10:03 PM 11.) 3y + 3x = - 6 12.) 3x = - 5y + 31 y = x + 16 x = 3y - 27 http://www.math.com/cgi-bin/mathf_sp.pl Page 2 of 2