# Solving Systems of Equations

### Math Models Lesson 11 By Dakota Grooms

## Introducing Solving Systems

Their are several ways of solving systems of equations. Graphing, Substitution, and Elimination. My preferred way to solve equations is graphing.

**Graphing** is when you put the two equations on a graph to find the point of intersection.

**Elimination** is canceling out everything to find the X and Y values.

**Substitution** is taking one equation and plugging it in the other to find the X value, then take that and plug it into the other equation to find the Y value

Any system can be solved by **graphing. **For example: plug in **y=4x+7** and **y=2/3x-2** in Y= on your calculator then press 2nd, trace, go down to intersect and press enter 3 times. The intersecting points are **(-2.7,-3.8)**. This is sometimes the easiest way to solve equations because its strait forward.

**REMEMBER: **Sometimes you have to write the equations out before you can solve them.

Equations can also be solved by **elimination** by canceling out everything to find the X and Y points.

For example: if you have the two equations **6****x+3y=12** and **-6x+y=-20** you would subtract them. Also, its a lot easier to solve by elimination if either the X or Y values cancel out.

**6x+3y=12**

**-6x+y=-20**

__4y=-8__

**4**

**Y=-2**

The X cancels out, you have 4 Y's and a -8. Divide them by 4 and **Y= -2**. Then plug in the Y value in one of the original equations to solve for X. The Y becomes a -6 and the equation is **6x-6=12**. Add 6 to both sides of the equals sign so its **6x=18**. After that divide by 6 and **X=3**.

**6x+3(-2)=12**

**6x-6=12**

__6x=18__

**6**

**X=3**

Another way to solve an equation is by substitution. For example: if you had the two equations **6x+8y=30** and **y=-4x+20 **you would plug in the second equation into the first equation's Y intercept and solve for X.** x=5**

**6x+8(-4x+20)=30**

**6x-32x+160=30**

**-26x+160=30**

__-26x=-130__

**-26**

**X=5**

Then plug in the X value into the second equation to find the Y value and **y=0**.

**y=-4(5)+20**

**y=-20+20**

**y=0**

## No Solution When two equations on a graph do not have a point of intersection | ## Point of Intersection Where two equations intersect on a graph to get the solution | ## Solution The X and Y values of two intersecting equations on a graph |