# 5.2 – Two-Variable Systems: Substitution

Key Terms

• Expression – A combination of numbers, variables, and operations that does not contain an equals sign or inequality sign.
• An expression that includes variables is often called an algebraic expression.
• Ex. 4 + 3x
• Substitution – The process of replacing a variable with a number or expression.
• Ex. 2x + 8, when you substitute x = 4: 2(4) + 8

Review

• Graphing is ONE way to locate the point where two lines meet (the solution)
• Ex. Graph both lines to see where they meet
• y = 2x + 1 (red line)
• y = -3x + 6 (blue line)
• They meet at point: (1, 3)

• Solutions
• If you get an identity, such as 0 = 0 or 6 = 6, the system has an infinite number of solutions.
• The lines are on top of one another, and have the same slope and the same y-intercept.
• If you get a contradiction, such as 0 = 7 or 6 = -6, then the system has no solution.
• The lines are parallel and will never cross.  They have the same slope and different y-intercepts.

Notes

• Substitution
• To solve a system of linear equations using the substitution method, you replace a variable with an equal value or expression.
• Write the answer as an ordered pair: (x, y).  x is always first, then y.

• Example

• Verifying the Solutions (from example above)
• There are two ways to verify that your solutions are correct
1. Graph the equations and see where they meet
1. Green Line: x = 2
2. Red Line: 2x +3y = 13
2. Substitute back into both equations
1. When you substitute y = 3 into the original equations, they both check out!

• Watch and copy these videos to have notes on a few more complicated examples

More Examples

• Ex 1.  Use the substitution method to solve the system of equations.
• y = 8x – 3
• x = 4
• Setup with Substitution: y = 8(4) – 3
• Evaluate: y = 12 – 3
• Simplify: y = 9
• Answer as an ordered pair (x, y): (4, 9)

• Ex 2. Use the substitution method to solve the system of equations.
• 2x + 4y = 14
• x = 3
• Setup with Substitution: 2(3) + 4y = 14
• Simplify: 6 + 4y = 14
• Subtract 6 on both sides: 4y = 8
• Divide by 4 on both sides: y = 2
• Answer as an ordered pair (x, y): (3, 2)

• Ex 3. Use the substitution method to solve the system of equations.
• y = 4x – 2
• y = x + 13
• Since both expressions equal y, set both equal to each other!
• Setup by Substitution: 4x – 2 = x + 13
• Add 2 to both sides: 4x = x + 15
• Subtract x on both sides: 3x = 15
• Divide by 3 on both sides: x = 5
• Substitute x = 5 back into either of the original equations:
• a. y = 4(5) – 2, which simplifies to y = 18
• b. y = 5 + 13, which also simplifies to y = 18
• Answer in ordered pair (x, y): (5, 18)

• Ex 4. Use the substitution method to solve the system of equations.
• 14x – 2y = 78
• 2x – 2y = 6
• Since we do not know what x or y equals, we need to choose an equation to simplify first.
• The 2nd equation has smaller numbers, and all are divisible by 2; so, let’s use that one:
• Divide all three terms by 2: $\frac{2x}{2}-\frac{2y}{2}=\frac{6}{2}$
• Simplify: x – y = 3
• Add y to both sides: x = 3 + y
• Substitute 3 + y for x in the 1st equation: 14(3 + y) – 2y = 78
• Distribute 14 into (3 + y): 42 + 14y – 2y = 78
• Combine like terms (y’s): 42 + 12y = 78
• Subtract 42 on both sides: 12y = 36
• Divide both sides by 12: y = 3
• Substitute y = 3 back in to either of the original equations.
• The 2nd one is easier, as the numbers are smaller: 2x – 2(3) = 6
• Simplify: 2x – 6 = 6
• Add 6 to both sides: 2x = 12
• Divide by 2 on both sides: x = 6
• Answer in ordered pair (x, y): (6, 3)
• You COULD also graph it and look for where the two lines meet