Warm-up

Twice the supplement of an angle is ten times the measure of the angle itself. Find the measure of both angles.

Three times the complement of an angle is seven times the measure of the angle itself. Find the measure of both angles.

Solving Systems of Linear Equations

What is a System of Linear Equations?

Wikipedia says it’s…

A collection of linear equations that use the same variables

What does it mean to be a solution?

It’s a coordinate pair that “works” in both equations

Graphically: It’s where the lines intersect

3 Different Methods

•Graphing•Substitution•Elimination/Linear Combination

Look at the Graph and Find the Solution to the system of equations

There are actually 3 cases

Case 1

Case 2

Case 3

One Solution

NO Solutions

Infinite Solutions

Ok so now let’s look at it algebraically

This is where the other two methods come in.

SubstitutionElimination/Linear Combination

SubstitutionWorks best when one of your coefficients is 1 (ex: x + 3y = 7,

x has a coefficient of 1 and y has one of 3.)First, get the variable with the coefficient of 1 alone (x = -3y +

7).Next, substitute the expression in for the variable you isolated

into the 2nd equation.Now you have an equation with one variable, so solve for that

variable. Then, plug your answer into an original equation to find the other variable.

So one of the equations (you choose which one) for one of the variables

(you choose which one), and then plugging this back into the other equation, "substituting" for the chosen variable and solving for the other. Then you back-solve for the first variable.

Example

2 3 2

4 24

x y

x y

Elimination/Linear Combination

First make one of the variables have opposite coefficients (if not already done) by multiplying.

Then you will add the equations together which will cancel out one of the variables and allow you to solve for the other.

Then you will plug it back into your equation and find the other variable.

Example

2 9

3 16

x y

x y

Try another…

2 9

3 4 14

x y

x y

How about some more!!

12 13 2

6 6.5 2

x y

x y

12 3 6

4 2

x y

x y

Flipped

Review

3 Different Methods

•Graphing•Substitution•Elimination/Linear Combination

Look at the Graph and Find the Solution to the system of equations

There are actually 3 cases

Case 1

Case 2

Case 3

One Solution

NO Solutions

Infinite Solutions

SubstitutionWorks best when one of your coefficients is 1 (ex: x + 3y = 7,

x has a coefficient of 1 and y has one of 3.)First, get the variable with the coefficient of 1 alone (x = -3y +

7).Next, substitute the expression in for the variable you isolated

into the 2nd equation.Now you have an equation with one variable, so solve for that

variable. Then, plug your answer into an original equation to find the other variable.

So one of the equations (you choose which one) for one of the variables

(you choose which one), and then plugging this back into the other equation, "substituting" for the chosen variable and solving for the other. Then you back-solve for the first variable.

Example

2 3 2

4 24

x y

x y

Elimination/Linear Combination

First make one of the variables have opposite coefficients (if not already done) by multiplying.

Then you will add the equations together which will cancel out one of the variables and allow you to solve for the other.

Then you will plug it back into your equation and find the other variable.

Example

2 9

3 16

x y

x y