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MATH121, Lesson 3.2 Notes

by: Mallory McClurg

MATH121, Lesson 3.2 Notes Math 121

Marketplace > University of Mississippi > Math > Math 121 > MATH121 Lesson 3 2 Notes
Mallory McClurg
GPA 3.37

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These notes cover all of Chapter 3.2, Linear Equations with 2 Variables. I give step-by-step explanations of every type of problem we may need to know how to solve, and I sketched out a graph for e...
College Algebra
Class Notes
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This 4 page Class Notes was uploaded by Mallory McClurg on Sunday September 18, 2016. The Class Notes belongs to Math 121 at University of Mississippi taught by Dirle in Fall 2016. Since its upload, it has received 6 views. For similar materials see College Algebra in Math at University of Mississippi.


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Date Created: 09/18/16
Math121 Chapter 3 Lesson 3.2 – Linear Equations in Two Variables EXAMPLE 1. -2x + 11(x + 2y) = 11x + y (When solving linear equations with two variables, we want to get it in the proper form ax + bx = c. If we cannot get it into this form, it is not linear! Simple enough, right? So first, distribute the 11 to the x and 2y.) -2x + 11x + 22y = 11x + y (Now combine like terms.) 9x + 22y = 11x + y (Now, subtract the terms with variables on the right side of the equation from the left side of the equation to get it in the proper form.) -2x + 21y = 0 (Yes! It’s linear. Yay.) EXAMPLE 2. (y + 3) – y = -5x + 1 (First, factor out the (y + 3) .) y + 6y + 9 – y = -5x + 1 (Now, combine like terms on like sides of the equation.) 6y + 9 = -5x + 1 (Subtract terms with variables on the right side of the equation from the left side of the equation.) 6y + 5x = -8 (Yes! It’s linear, because this is in proper form.) EXAMPLE 3. 2 2 2 8x – (x + 1) = y – 2 (First, factor out the (x + 1) .) 8x – (x +2x + 1) = y – 2 (Take away the parentheses from the quadratic, and change the signs to simplify the lef2 side of the equation.) 7x – 2x – 1 = y – 2 (Subtract terms with variables on the right side of the equation form the left side of the equation.) 2 7x – 2x – y + 1 = 0 (This is not in proper form; NOT LINEAR.) EXAMPLE 4. x + 3xy = 4y (Can we simplify this to put it in proper form? No….. NOT LINEAR.) EXAMPLE 5. (11/x) + (7/y) = 6 (Can we simplify this to put it in proper form? No… NOT LINEAR.) Math121 Chapter 3 EXAMPLE 6. (11x – y)/2 + 3y = 8 (Separate the (11x – y)/2 into two fractions.) (11x/2) – (y/2) + 3y = 8 (To combine the terms on the left side of the equation with like variables, we need to turn 3y into a fraction with 2 as the denominator. So multiply the fraction (3y/1) by (2/2).) (11x/2) – (y/2) + (6y/2) = 8 (Now, combine fractions with like terms and put it in proper ax + by = c form.) (11/2)x – (5/2)y = 8 (This is in proper LINEAR form!) EXAMPLE 7. 3x + 7y = -21 (We are asked to find all x and y intercepts in this linear equation. Since we know that an x intercept has a y value of y = 0, and a y intercept has an x value of x = 0, we can find each intercept by writing two equations, with x = 0 in one, and y = 0 in the other.) 3(0) + 7y = -21 3x + 7(0) = -21 7y = -21 3x = -21 y = -3 x = -7 (0, -3) (-7, 0) EXAMPLE 8. 5y – 4x = -20 (To find the intercepts, let y = 0 in one equation and x = 0 in the other, because we know they intercept the axes where the value is zero.) 5(0) – 4x = -20 5y – 4(0) = -20 -4x = -20 5y = -20 x = 5 y = -4 (5, 0) (0, -4) EXAMPLE 9. -2y + 3x = -2y + 6 (We need to put this in proper ax + by = c form, so add 2y to both sides of the equation.) Math121 Chapter 3 3x = 6 (The terms with y as a variable both cancelled each other out…, so the y intercept value is absent.) x = 2 (2, 0) (y is absent) EXAMPLE 10. -6y = - 48 (The x variable is absent, so solve for the value of y.) y = 8 (0, 8) (x is absent) EXAMPLE 11. -3x = -3y (If we divide both sides by -3, we see that it’s y = x.) x = y (In this situation, we should realize that this means x = 0 (0, 0) and y = 0 (0, 0).) Math121 Chapter 3 EXAMPLE 12. 8x + 14 = -8y + 14 (Subtract 14 from both sides.) 8x = -8y (Divide both sides by -8.) -x = y EXAMPLE 13. 7y – 7 = 2x (Get this in ax + by = c form.) -2x + 7y = 7 (In two separate equations, let x = 0 in one, and y = 0 in the other to find the intercepts.) -2(0) + 7y = 7 -2x + 7(0) = 7 7y = 7 -2x = 7 y = 1 x = (-7/2) (0, 1) (-7/2, 0) EXAMPLE 14. 8 – 2y = 28 – 5x (Write out into two equations, trying to get the y on a side by itself in one, and the x on a side by itself in the other.) -2y = 20 – 5x -20 – 2y = -5x y = -10 – (5/-2)x x = 4 – (2/-5)y (Now, set each equal to zero to find the x and y intercepts.) (5/2)x – 10 = 0 (2/5)y + 4 = 0 10 = (5/2)x -4 = (2/5)y x = 4 y = -10 (4, 0) (0, -10)


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