**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)

- Ex. Graph both lines to see where they meet

- 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.

- If you get an identity, such as 0 = 0 or 6 = 6, the system has an infinite number of solutions.

**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

Answer: (2, 3)

- Verifying the Solutions (from example above)
- There are two ways to verify that your solutions are correct
- Graph the equations and see where they meet
- Green Line: x = 2
- Red Line: 2x +3y = 13

- Substitute back into both equations
- When you substitute y = 3 into the original equations, they both check out!

- Graph the equations and see where they meet

- There are two ways to verify that your solutions are correct

- 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:
- 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

- Since we do not know what x or y equals, we need to choose an equation to simplify first.