Linear combination
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This document provides instructions on how to solve systems of linear equations using the linear combination method, also known as the elimination method or add/subtract method. It gives examples of using this method to solve systems by eliminating a variable through adding or subtracting the equations. Practice problems are provided for the reader to solve systems of equations using the linear combination method.
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This document contains terminology and examples related to functions. It defines f(x) as "the value of y in function f, where the independent variable is x". It provides examples of evaluating functions like f(x) = -7x + 100 for different x values. It also gives examples of composite functions like f(f(x)) and using different functions like f(x) and g(x) together. The examples relate to word problems about distances over time on a boat trip.
Translating words into symbols
This document provides examples of translating word problems into mathematical equations and then solving those equations. It contains sample word problems that translate to equations involving addition, subtraction, multiplication and division. It also includes guidance on common phrases like "less than" and their symbolic representations. Finally, it provides two word problems as examples for students to practice translating words into mathematical symbols.
Inequalities
This document provides information and examples about solving inequalities:
1) It gives examples of solving linear equations with one variable and two variables.
2) It explains the symbols used for different inequality types and how to graph them, including shading the correct side of the inequality.
3) It provides rules for solving inequalities, such as reversing the inequality sign when multiplying or dividing both sides by a negative number.
4) Multiple examples are worked through step-by-step of solving and graphing various inequalities with one variable and multiple variables.
Solving equations with variables on both sides & translating words into symbols
This document contains sample math problems and notes about solving equations with variables on both sides. It includes examples of solving equations like 5y - 3y + 13 = 26, 3s + 2 = 8, 2d + 3d = 15, and 2(w -1) + 8 = 16. The notes provide instructions for solving equations where variables are on both sides, such as 6x = 4x + 18, 3y = 15 - 2y, and 7(a - 2) - 6 = 2a + 8 + a. The document also references a Solving Equations BINGO activity and exit slip homework.
4.3 cramer’s rule
The document provides information about solving systems of linear equations using Cramer's rule, including:
- Cramer's rule uses determinants of the coefficient matrix to solve systems of linear equations.
- For a 2x2 system, if the determinant of the coefficient matrix is not equal to 0, there is exactly one solution.
- The solutions are found by taking the determinants of the numerator matrices (with one column replaced by the constants) divided by the determinant of the coefficient matrix.
- Several examples demonstrate solving 2x2 systems using Cramer's rule.
Solving equations
This document provides an introduction to solving equations. It explains the key properties of equality that must be followed when solving equations, such as performing the same operation to both sides of the equal sign. Examples are provided to demonstrate applying these properties to solve one-step, two-step, and multi-step equations. Students are instructed to show their work and given practice problems applying the steps to solve different types of equations.
Transforming equations
This document provides background information on transforming equations and solving one-step equations using addition and subtraction. It explains that if the same number is added to or subtracted from both sides of an equation, the resulting equations will be equal. Examples are provided of solving equations by adding or subtracting the same number from both sides to isolate the variable.
Solving percent equations
The document discusses percentages and solving percentage equations. It provides examples of how to write percentage equations for word problems, such as finding what percentage one number is of another or finding the total given a percentage. It demonstrates solving equations to find percentages and totals in various contexts like test scores, class sizes, sales, and more. Percentages allow for easier comparison of quantities and are used in many everyday situations like grades, weather, nutrition labels, and shopping.
Decimals, fractions, and percents
This document discusses how to convert between decimals, fractions, and percents. It provides examples for converting non-repeating and repeating decimals to fractions by dropping the decimal point and putting the number over the place value. It also demonstrates how to convert fractions to decimals by dividing the numerator by the denominator and moving the decimal point. Additionally, it shows how to convert decimals to percents by moving the decimal point two places to the right and adding the percent sign, and how to convert percents to decimals by moving the decimal point two places to the left and dropping the percent sign.
Notes on 3.2 properties of linear frunction graphs
The document discusses key properties of linear function graphs, including that the y-intercept is the y-value when x=0 and the x-intercept is the x-value when y=0. It defines slope as m in the slope-intercept equation y=mx+b and describes horizontal and vertical lines having slopes of 0 and undefined respectively. It lists three common forms for linear functions: slope-intercept, point-slope, and Ax+By=C and provides examples of transforming between the forms.
Proportions & school pictures
The document discusses proper proportions for various items like wallets, photos, and the American flag. It poses a math problem asking what the width of a 57cm long American flag would need to be in order to have the official proportion of 19 units in length for every 10 units in width.
Inequalities
This document discusses inequalities and provides examples of solving various inequality expressions. It begins by giving an example of a negative number that is not closed under multiplication. It then defines common inequality symbols like less than, greater than, less than or equal to, and greater than or equal to. The document proceeds to give examples of solving linear inequalities assuming the domain is all real numbers, including compound inequalities involving absolute value expressions. It concludes with examples for an exit slip problem set involving solving several inequalities.
The matrix
The document contains information about the reciprocal of 1/9, the value of |-8+-2|, the difference between a January wind chill of -8 degrees and an August heat index of 108 degrees. It also contains a summary of the 1999 film The Matrix and definitions of a matrix and rules for adding, subtracting and multiplying matrices by scalars.
Simplifying expressions, solving, domain and range
This document appears to be notes from a math class covering simplifying expressions, solving equations, and domain and range. It includes examples of irrational and imaginary numbers, naming polynomials by degree and terms, multiplying binomials, homework assignments on the topics, and notes on using the vertical line test to determine if a relation is a function by checking if each x-value has exactly one y-value. Sketching graphs of functions given domains and ranges is also mentioned.
04.07.10 more volume exercises
The document provides instructions to sketch and calculate the volumes of different shapes, including a rectangular prism with dimensions of 3 mm by 5 mm by 15 mm, a cylinder with a diameter of 2.6 m and height of 3.5 m, a rectangular prism with dimensions of 6 inches by 6 inches by 9 inches, and a cylinder with a radius of 3 yards and height of 10 yards. It also gives problems to find the height of a rectangular prism with given length, width, and volume, and to find the radius and height of cylinders with given dimensions and volumes.
04.08.10 applications of volume and surface area
The document contains 4 applications that use surface area and volume concepts:
1) Finding the height of an aquarium tank given its dimensions and volume.
2) Comparing the surface area and volume of a can and box to determine which holds more juice.
3) Calculating the surface area of a gift to see if a sheet of wrapping paper is large enough.
4) Determining which pizza sauce jar is a better buy based on price and volume.
04.07.10 more volume
The document provides instructions for calculating the volumes of different shapes, including a rectangular prism and cylinder. It asks the reader to first predict which solids have the greatest and least volumes without calculations. It then has the reader measure dimensions of different solids to calculate their actual volumes and rank them. Finally, it gives two word problems to find heights using given lengths, widths, radii and volumes.
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Solving equations with variables on both sides & translating words into symbols
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みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
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みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
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Linear combination
- 1. ACT OpenerACT Opener Solve each system by substitution. 1) 2) 3) 4)
- 2. Solve System bySolve System by Linear Combination MethodLinear Combination Method 1. Determine which variable to eliminate. 2. Determine by what number to multiply each equation. 3. Multiply each equation. 4. Determine whether to add or subtract, then do it. – Add if the coefficients are opposites. – Subtract if the coefficients are the same. 1. Substitute to find the other term.
- 3. Linear Combination MethodLinear Combination Method • This method is also known as – Elimination – Add/Subtract Method
- 4. Example 1Example 1 x – y = 7 x + y = 3 2x + 0y = 10 2x = 10 x = 5 x + y = 3 5 + y = 3 y = -2 (5,-2) Solve using the addition method.
- 5. Example 2Example 2 2x + 3y = 11 -2x + 9y = 1 0x + 12y = 12 12y = 12 y = 1 2x + 3y = 11 2x + 3(1) = 11 2x + 3 = 11 2x = 8 x = 4 (4,1) Solve using the addition method.
- 6. PracticePractice 1) Solve using the addition method. x y 5 2x y 4 + = − = 2) 3x 3y 6 3x 3y 0 − = + =
- 7. Example 3Example 3 Solve using the addition method. 3x – y = 8 x + 2y = 5 x + 2y = 5 3x – y = 8 multiply as needed(-3)( ) ( )(-3) 3x – y = 8 -3x – 6y = -15 addition property -7y = -7 y = 1 3x - (1) = 8 3x = 9 x = 3 (3,1)
- 8. Example 4Example 4 8x 2y 10 4x 15 3y + = − = + 4x – 3y = 15 write in standard form8x + 2y = -10 multiply as needed(-2)( ) ( )(-2) 8x + 2y = -10 -8x + 6y = -30 addition property 8y = -40 y = -5 8x + 2(-5) = -10 8x - 10 = -10 8x = 0 x = 0 (0,-5) Solve using the addition method.
- 9. PracticePractice 1) Solve using the addition method. 5x 3y 17 5x 2y 3 + = − = − 2) 8x 11y 37 11y 7 2x + = − =
- 10. PracticePractice 1) Solve using the addition method. 5x 3y 17 5x 2y 3 + = − = − 2) 8x 11y 37 11y 7 2x + = − =