Substitution involves replacing one term for another in an equation or system of equations. To solve a system using substitution, one equation is solved for one variable and substituted into the other equation. This creates a single variable equation that can then be solved. Parallel lines yield a false statement when substituted into each other, indicating they have no solution, while the same line yields a true statement and infinite solutions. When choosing a variable to substitute, select the one without a coefficient to more easily substitute terms.

1. Solving Systems With Substitution

2. Substitution What is substitution? The replacement of one thing for another If a = 5 and a + b = 8 then what’s b? 3

3. Substitution in a system y = 2x + 2 y = x + 3 Choose one of the equations to substitute into the other equation. Let’s substitute the second equation into the first equation.

4. Substitution in a system y = 2x + 2 y = x + 3 x +3 = 2x + 2 Now solve for x x = 1

5. Substitution in a system y = 2x + 2 y = x + 3 Now substitute the 1 into either equation for the x and solve for the y. y = 2(1) + 2 y = 4 (1, 4) is the solution

6. Parallel Lines 3x + y = 5 6x + 2y = 1 Solve the first one for y to substitute into the second equation. y = -3x + 5 6x + 2( -3x + 5) = 1

7. Parallel Lines 6x + 2( -3x + 5) = 1 6x – 6x + 10 = 1 10 = 1 FALSE STATEMENT No solution – lines are parallel

8. Same Lines 3x - y = -2 y = 3x + 2 Substitute second one into the first equation. 3x – ( 3x + 2) = -2 3x – 3x – 2 = - 2 - 2 = - 2 TRUE STATEMENT Infinite number of solutions! Same lines.

9. Choosing a Variable How do you know what variable to substitute for? x + 3y = 12 3x + 9y = 8 Go with the variable that doesn’t have a coefficient! Which one is that? x = -3y + 12

10. Choosing a Variable x = -3y + 12 3x + 9y = 8 3 ( -3y + 12) + 9y = 8 Then solve from there