Systems of Equations Lesson

Solving Systems of Equations: Graphing

More than one equation

Systems of Equations



-A system of equations is any set of equations in the same plane.

-To solve a system of equations means to find the points that make all of the equations in the system true.

-When you have two variables (x and y) you need two equations to solve the system.

Solve by graphing

When you graph the two equations you will have three outcomes.

Intersecting - one solution

Parallel - no solution

Same line (coinciding) - infinite solutions

Multiple Choice

How many solutions?

• One Solution
• No solution
• Infinitely Many Solutions

Multiple Choice

If a system of linear equations has one solution, what does this mean about the two lines? 

• Parallel lines

• the same line 

• Intersecting lines

Multiple Choice

How many solutions does this system of equations have? (Yes, there are two lines.)

• One Solution

• No solution

• Infinitely Many Solutions

• Two Solutions

One Solution

When lines intersect (cross) there is one solution. That solution is the coordinate where the lines intersect each other.

Given

y=-x+7

y=2x+1

x=2 and y=5

Multiple Choice

What is the solution to the system?

• (0, 3)

• (1, -1)

• (-3, 1)

• (1, 3)

Multiple Choice

How many solutions will this system have?

• One

• Two

• No Solution

• Infinitely Many Solutions

Multiple Choice

^{What is the solution?} 

• 1

• -2

• (1, 2)

• (1, -1)

Multiple Choice

Steffen graphed two lines in order to find the solution to a given system of equations.
What is the solution?

• (-3,-8)

• (-8,-3)

• (3,-8)

• (8,3)

Multiple Choice

What is the solution?

• (1, -1)

• (-1, 1)

• (0, -2)

• (0, 1)

Review Slope Intercept Form

y = mx + b

To graph slope intercept form begin with b (the y-intercept) and then rise and run from that point to find the next point on the line.

Multiple Choice

Which is the graph of the equation y = -2x + 1?

Multiple Choice

Which graph best represents the equation y = -3x + 4?

Multiple Choice

Which could be the graph of the equation y = 1/3x - 2?

Graphing standard form

The easiest way to graph standard form is to graph the intercepts.

x-intercept

substitute y=0 solve for x

y-intercept

x=0 solve for y

Multiple Choice

Which equation is written in STANDARD form?

• y = 1/2x + 2

• y - 3 = -5(x + 2)

• -7x + 2y = 9

• 3x + 2 = y

Multiple Choice

Which graph represents x + y = 4

Multiple Choice

Which graph represents y = 4 + 2x?

Multiple Choice

Which graph represents 2x - 5y = 1?

Multiple Choice

Which graph represents -x + 2y = 3?

Multiple Choice

Which graph represents 2x + 4y = 12?

Systems of Equations

​Solving Algebraically

Solving by Substitution

Steps 1 & 2

Solving by Substitution

Steps 3 & 4

The solution is (42,-12)

Multiple Choice

Solve the following system:

$$y=-2$$  

$$4x-3y=18$$  

• $$\left(3,\ -2\right)$$  

• $$\left(3,\ 2\right)$$  

• $$\left(-2,3\right)$$  

• $$\left(2,\ 3\right)$$  

Multiple Choice

Solve the following systems of equations using substitution:

x = 6

y = 2x - 3

• (6, 6)

• (6, 9)

• (9, 6)

• (9, 9)

Multiple Choice

Solve this system by substitution.

• (1, -4)

• (1, 4)

• (-4, -7)

• (-7, -4)

Multiple Choice

Solve the systems by substitution:

x = 4

-2x + 3y = -2

• (4, 6)

• (4, 4)

• (4, 2)

• 2, 4)

Solving by Elimination

Steps 1 & 2

Solving by Elimination

Steps 3/4 & 5/6

The solution is (2,4)

Multiple Choice

3. What should be the first step to solve the system of equations 2+5y=17 and 6x-5y=-9?

• Add teo equations to eliminate y.

• Subtract two equations to eliminate y.

• Add two equations to eliminate x

• Subtract two equations to eliminate x.

Multiple Choice

4. What is the solution of the system 5x-6y=-32 and 3x+6y=48?

• (7, 2)

• (-7,2)

• (2,7)

• (-2,7)

Multiple Choice

5. What is the solution of the system of equations -3x-3y=9 and 3x-4y=5?

• (2, 1)

• (-2, -1)

• (1,2)

• (-1,-2)

Multiple Choice

Solve the following system by using elimination:

3x - 5y = 8

2x + 5y = -3

• (1, 5)

• (1, -1)

• (-1, -1)

Multiple Choice

Solve using Elimination. 
 4x + 8y =  20
-4x + 2y = -30

• (-7,1)
• (2,-5)
• (-2,5)
• (7,-1)

Multiple Choice

Lesly graphed two lines in order to find the solution to a given system of equations.
What is the solution?

• (-3,-8)
• (-8,-3)
• (3,-8)
• (8,3)

Multiple Choice

Solve the following system by graphing.  What is the solution?

• Infinite number of solutions
• (3, 3)
• (3, -3)
• (-3, 3)

Multiple Choice

If a system of linear equations has one solution, what does this mean about the two lines? 

• Parallel lines
• the same line 
• Intersecting lines