Review: Solve Equations & Intro: Systems

Review: Solve Equations
&
Intro: Systems

Multiple Choice

3x - 5 = 10

What is the first step to solving this equation?

• add 5

• subtract 5

• divide 5

• multiply 5

Multiple Choice

3x = 15

What is the next step to solving this equation?

• add 3

• subtract 3

• divide 3

• multiply 3

Multiple Choice

$$\frac{x}{5}=5$$

What is the first step to solving this equation?

• add 5

• subtract 5

• divide 5

• multiply 5

Multiple Choice

$$\frac{x}{5}=5$$

• $$x=1$$

• x=25

• x=0

• x=10

Multiple Choice

$$\frac{x}{2}=4$$

What is the first step to solving this equation?

• add 2

• subtract 2

• divide 2

• multiply 2

Multiple Choice

$$\frac{x}{2}=4$$

• x=2

• x=6

• x=8

• x=-2

Multiple Choice

4x=16

What is the first step to solving this equation?

• add 4

• subtract 4

• divide 4

• multiply 4

Multiple Choice

4x=16

• x=12

• x=20

• x=64

• x=4

Multiple Choice

5 = 4x - 15

What is the first step to solving this equation?

• add 15

• subtract 15

• divide 15

• multiply 15

Multiple Choice

5 = 4x - 15

The second step to solving this equation is?

• add 4

• subtract 4

• divide 4

• multiply 4

Multiple Choice

5 = 4x - 15

• x=16

• x=-15

• x=5

• x=4

Multiple Choice

7x + 8 = 29

What is the first step to solving this equation?

• add 8

• subtract 8

• divide 8

• multiply 8

Multiple Choice

7x + 8 = 29

The second step to solving this equation is?

• add 7

• subtract 7

• divide 7

• multiply 7

Multiple Choice

7x + 8 = 29

• x=3

• x=-3

• x=7

• x=-7

Multiple Choice

Solve for x:

 

$$\frac{x}{2}=1$$  

• $$x\ =\ 2$$  

• $$x\ =\ 1$$  

• $$x\ =\ \frac{1}{2}$$  

• $$x\ =\ -2$$  

Multiple Choice

Solve for x:

 

$$5x=30$$  

• $$x\ =\ \frac{1}{6}$$  

• $$x\ =\ 150$$  

• $$x\ =\ 6$$  

• $$x\ =\ -6$$  

Multiple Choice

Solve for x:

 

$$x+2=10$$  

• $$x\ =\ 12$$  

• $$x\ =\ 8$$  

• $$x\ =\ -8$$  

• $$x\ =\ -12$$  

Introduction to Solving Systems of Equations
using:

Graphing/Substitution/Elimination

Three Methods for Solving Systems

Which method you use depends on what form(s) your equations are in.

Solving Systems By Graphing

This method works best with equations in slope-intercept form, when you have graph paper or a graphing calculator.

What is a System of Equations?

It is simply two or more equations.

On the graph is ...

y = 2x + 1

y = -x + 7

To solve a system, you find the point where the lines intersect.

On this graph it is (2,5)

The SOLUTION (2,5) makes each equation true.

Plug in to each equation.

y = 2x + 1

5 = 2(2) + 1

5 = 4 + 1

5 = 5

y = -x + 7

5 = -2 + 7

5 = 5

Multiple Choice

What is the solution to this system of equations?

• (-1,4)

• (4,1)

• (4,-1)

• (1,4)

Answer is: (4,-1)

Find the answer at the point where the lines cross.

Why not always graph?

As you can see in the graph, sometimes where the lines cross is not at a perfect point.

So other methods are needed to be get an accurate answer.

We will only work with two equations

There is more than one way to solve systems.

The one I just showed was graphically.

Now we are going to focus on the method of SUBSTITUTION

Solving Systems Using Substitution

Use this method when one or both equations are or are easily solved for one of the variables.

Video

This video shows the situation where y is by itself in both equations.

Step 1: One of your equations is already set up as x=_____ or y=_____.

In fact, they both are.​

​

Step 2: Substitute y = -4x + 8 into the other equation for y.​

Step 2 continued: solve for x.

​

Step 3: Substitute x = 3 into one of your original equations and solve for y.

​

Step 4: Write your answer as (3, -4)​.

​

​

​

Solve the System

y = x - 3

y = -x + 5

Since BOTH equal y ... we can substitute one of the expressions for y.

y = x - 3

y = -x + 5

Substitute

x - 3 = -x + 5

Solve: x - 3 = -x + 5

You can see the work to the right.

We get x by itself.

BUT we are not done.

The answer is where the lines cross.

So we need a coordinate. (x,y)

We now have the x.

We need to find the y.

Original System

y = x - 3

y = -x + 5

Our answer from the substitution...

x = 4

We can pick EITHER equation, plug in the x and find the y.

Let's use: y = x - 3

(you can see the work to the right)

Answer is: (4,1)

Solve the System

y = -x - 1

y = -5x - 17

Since BOTH equal y ... we can substitute one of the expressions for y.

y = -x - 1

y = -5x - 17

Substitute

-x - 1 = -5x - 17

Multiple Choice

Solve the following system:

$$y=-x-1$$  

$$y=-5x-17$$  

• $$\left(-4,3\right)$$  

• $$\left(4,\ 3\right)$$  

• $$\left(3,\ 4\right)$$  

• $$\left(3,\ -4\right)$$  

Solve the System

y = x - 1

y = 2x + 2

Since BOTH equal y ... we can substitute one of the expressions for y.

y = x - 1

y = 2x + 2

Substitute

x - 1 = 2x + 2

Answer is: (-3,-4)

Multiple Choice

Solve the following system:

$$y=x-1$$  

$$y=2x+2$$  

• $$\left(-4,-3\right)$$  

• $$\left(4,\ 3\right)$$  

• $$\left(3,\ 4\right)$$  

• $$\left(-3,\ -4\right)$$  

Video

This video shows the situation where y is by itself in just one equation.

Solve the System

y= 4x - 11

-4x + 3y = -1

Since y is by itself in ONLY ONE equation ... we must substitute what y equals into the other equation in place of the y.

y = 4x - 11

-4x + 3y = -1

Substitute

-4x + 3(4x - 11) = -1

Answer is: (4,5)

Multiple Choice

Solve the following system:

$$y=4x-11$$  

$$-4x+3y=-1$$  

• $$\left(4,-5\right)$$  

• $$\left(-4,\ 5\right)$$  

• $$\left(4,5\right)$$  

• $$\left(5,-4\right)$$  

• $$\left(-5,4\right)$$  

Solve the System

y= x - 1

2x - 3y = -1

1) Solve for x

2) Plug in the x and solve for y

ANSWER (4,3)

Multiple Choice

Solve the following system:

$$y=x-1$$  

$$2x-3y=-1$$  

• $$(-4,3)$$  

• $$(4,3)$$  

• $$(-4,-3)$$  

• $$\left(4,-3\right)$$  

Solve Systems Using Elimination

Use this method when your equations are or are easily written in the same format. This is easiest when one set of terms "matches".

Video

This video shows the Elimination Method.