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Solving Systems of Equations by Substitution: A Comprehensive Guide

Learn how to solve systems of equations using the substitution method. This guide details the steps to rewrite equations, substitute variables, and solve for unknowns. Two practical examples illustrate the process: solving the equations 4x + 3y = 4 and 2x - y = 7, as well as x + 6y = 2 and 5x + 4y = 36. Follow along to develop a strong foundation in algebraic systems. Check your answers for accuracy and practice with assigned homework problems to reinforce your understanding.

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Solving Systems of Equations by Substitution: A Comprehensive Guide

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  1. p.123 #55 SOLUTIONS • p = independent, n = dependent b) n = -1600p + 14,800 A: ($8, 2000 widgets) B: ($3, 10000 widgets) c) n = -6000p + 32,000

  2. 3.2 Solving Systems Algebraically Solving Systems by Substitution

  3. 1) Solving Systems by Substitution • Sometimes graphing a system of equations produces a solution that is difficult to interpret

  4. 1) Solving Systems by Substitution • Sometimes it’s easier to use algebra to solve a system of equations • How??

  5. 1) Solving Systems by Substitution Example 1: Solve the system by substitution. 4x + 3y = 4 2x – y = 7 {

  6. 1) Solving Systems by Substitution Example 1: Solve the system by substitution. 4x + 3y = 4 2x – y = 7 { Substitution – “sub” one equation into the other

  7. 1) Solving Systems by Substitution Example 1: Solve the system by substitution. 4x + 3y = 4 2x – y = 7 { Substitution – “sub” one equation into the other • Number the equations as 1 and 2 • Re-write one equation as x = OR y = • Sub one equation into the other. Solve for the unknown. • Sub the known value into the other equation. Solve for the remaining unknown.

  8. 1) Solving Systems by Substitution Example 1: Solve the system by substitution. 4x + 3y = 4 1 2x – y = 7 2 Step 1: Number the equations {

  9. 1) Solving Systems by Substitution Example 1: Solve the system by substitution. 4x + 3y = 4 1 2x – y = 7 2 Step 2: Re-write one of the equations as x= ORy=, whichever is easiest {

  10. 1) Solving Systems by Substitution Example 1: Solve the system by substitution. 4x + 3y = 4 1 2x – y = 7 2 2x – 7 = y y = 2x – 7 Step 2: Re-write one of the equations as x= ORy=, whichever is easiest {

  11. 1) Solving Systems by Substitution Example 1: Solve the system by substitution. 4x + 3y = 4 1 2x – y = 7 2 2x – 7 = y y = 2x – 7 Step 3: Sub y = 2x – 7into equation 1 . Solve for x. {

  12. 1) Solving Systems by Substitution Example 1: Solve the system by substitution. 4x + 3y = 4 1 4x + 3(2x– 7) = 4 Step 3: Sub y = 2x – 7into equation 1 . Solve for x. “y” becomes “2x – 7”

  13. 1) Solving Systems by Substitution Example 1: Solve the system by substitution. 4x + 3y = 4 1 4x + 3(2x– 7) = 4 4x + 6x – 21 = 4 10x = 21 + 4 10x = 25 x = 2.5 “y” becomes “2x – 7”

  14. 1) Solving Systems by Substitution Example 1: Solve the system by substitution. 4x + 3y = 4 1 4x + 3(2x– 7) = 4 4x + 6x – 21 = 4 10x = 21 + 4 10x = 25 x = 2.5 “y” becomes “2x – 7” Step 4: Sub x = 2.5into equation 2 . Solve fory.

  15. 1) Solving Systems by Substitution Example 1: Solve the system by substitution. y = 2x – 7 2 y = 2(2.5) – 7 y = -2 “x” becomes “2.5” Step 4: Sub x = 2.5into equation 2 . Solve fory.

  16. 1) Solving Systems by Substitution Example 1: Solve the system by substitution. Therefore, the solution is (2.5, -2).

  17. 1) Solving Systems by Substitution Example 2: Solve the system by substitution. Check your answer. x + 6y = 2 5x + 4y = 36 {

  18. 1 2 1) Solving Systems by Substitution Example 2: Solve the system by substitution. Check your answer. x + 6y = 2 x = 2 – 6y 5x + 4y = 36 1 {

  19. 1 1 2 2 1) Solving Systems by Substitution Example 2: Solve the system by substitution. Check your answer. x + 6y = 2 x = 2 – 6y Sub in 5x + 4y = 36 5(2– 6y) + 4y = 36 10 – 30y + 4y = 36 -26y = 26 y = -1 {

  20. 1 1 1 2 2 1) Solving Systems by Substitution Example 2: Solve the system by substitution. Check your answer. x + 6y = 2 x = 2 – 6y Sub in 5x + 4y = 36 5(2– 6y) + 4y = 36 10 – 30y + 4y = 36 -26y = 26 y = -1 Sub in {

  21. 1 2 1) Solving Systems by Substitution Example 2: Solve the system by substitution. Check your answer. x + 6y = 2 5x + 4y = 36 x = 2 – 6(-1) x = 8 {

  22. 1) Solving Systems by Substitution Example 2: Solve the system by substitution. Check your answer. x + 6y = 2 5x + 4y = 36 Therefore, the solution to the system is (8, -1). {

  23. 1) Solving Systems by Substitution Example 2: Solve the system by substitution. Check your answer. x + 6y = 2 5x + 4y = 36 Check: 8 + 6(-1) = 2 5(8) + 4(-1) = 36 2 = 2 36 = 36 {

  24. Homework p.128 #1-5, 13, 47, 48, 52 Tomorrow: Solving Systems by Elimination

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