Systems of Equations

Systems of Equations

by Krista Simonsen

When 2 lines are drawn on the same graph, there are 3 possible outcomes.

• The lines intersect

• The lines don't intersect

• The lines are on top of each other

Multiple Choice

Here is an example of 2 lines being graphed. What answer choice matches the situation shown?

• The liness intersect

• The lines do not intersect

• The lines are on top of each others

For lines not to intersect at all...

• The lines must be parallel

• If the lines are parallel, we can write 'NO SOLUTION'

• Parallel lines have the SAME SLOPE but DIFFERENT y-intercepts

Multiple Choice

Here is an example of 2 lines being graphed. What answer choice matches the situation shown?

• The liness intersect

• The lines do not intersect

• The lines are on top of each others

For lines to be right on top of one another...

• You should only see one line even though there are 2 equations.

• When this situation occurs, you can write "INFINITE SOLUTIONS"

• The 2 equations have to be mathematically the equal to one another for the lines to be on top of one another.

Multiple Choice

Here is an example of 2 lines being graphed. What answer choice matches the situation shown?

• The liness intersect

• The lines do not intersect

• The lines are on top of each others

For lines to intersect once...

• The place where the line intersects is at an ordered pair

• Write the orderd pair in the form of (x,y).

• The slopes of the two lines must be different.

Multiple Choice

What is the point of intersection for the problem at the left, if any?

• (4,2)

• (-2,1)

• No Solution

• Infinite Solutions

Multiple Choice

What is the point of intersection for the problem at the left, if any?

• (4,2)

• (-2,1)

• No Solution

• Infinite Solutions

Multiple Choice

What is the point of intersection for the problem at the left, if any?

• (4,2)

• (-2,1)

• No Solution

• Infinite Solutions

Multiple Choice

What is the point of intersection for the problem at the left, if any?

• (1,-4)

• (-1,-4)

• (-4,1)

• (-4,-1)

Solving Systems of Linear Equations by Substitution

Step #1 Isolate a variable in one of the equations to get an expression

Solving Systems

Steps #2 Substitute that expression (from step #1) into the other equation.

Solving Systems

Step #3 Solve for the intersection point (x , y).

Solving Systems

step #4 write your answer as an order pair (x, y).

Multiple Choice

The solution to a system of equations is any ordered pair that makes both equations true. 

• TRUE

• FALSE

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 this system by substitution.

• (1, -4)

• (1, 4)

• (-4, -7)

• (-7, -4)

Multiple Choice

Solve the systems by substitution:

x = -4y

x - y = 15

• (-3, 12)

• (12, 3)

• (3, 12)

• (12, -3)

Multiple Choice

Which of the following correctly shows substituting the first equation into the second equation?

• y = 9(2x + 10) - 2

• 9x - 2 = 3x + 10

Multiple Choice

Which of the following correctly shows substituting the first equation into the second equation?

• 5x = -x + 4

• 5(y + 4) + 6y = 13

• 5x + 6(-x + 4) = 13

• 5x + (-x + 4) = 13

Multiple Choice

Which of answer shows the correct steps to solve for x?

• $$-3x+4\left(3x-5\right)=7$$ $$-3x+12x-5=7$$ $$9x-5=7$$ $$9x=12$$ $$x=1.25$$

• $$-3\left(3x-5\right)+4x=7$$ $$-9x+15+4x=7$$ $$-5x+15=7$$ $$-5x=-8$$ $$x=1.6$$

• $$-3x+4\left(3x-5\right)=7$$ $$-3x+12x-20=7$$ $$9x-20=7$$ $$9x=27$$ $$x=3$$

• $$-3x+4\left(3x-5\right)=7$$ $$-3x+12x-15=7$$ $$9x-15=7$$ $$9x=9$$ $$x=1$$

***VERY IMPORTANT SLIDE***

Multiple Choice

I can combine 2 linear equations if the coefficients are not opposites.

• True

• False

SYSTEMS OF EQUATIONS ARE 2 EQUATIONS WITH 2 VARIABLES

One way to solve them is by ELIMINATING one of the variables

Multiple Choice

Which variable has opposite coefficients?

• X

• Y

Multiple Choice

Which variable can we ELIMINATE?

$$5x\ +\ 8y\ =\ 23$$   $$-5x\ +\ 4y\ =\ 5$$  

• X

• Y

WHEN SOLVING USING ELIMINATION

• Look for opposite coefficients

• IF THERE IS NONE, MULTIPLY EVERY TERM IN AN EQUATION

Multiple Choice

Solve by elimination  -2x + 6y = 16 -4x  - 3y = 2

• (2,2)

• (-2, -2) 

• (-2,2)

• (2,1)

Multiple Choice

Solve for x and y 3x + 2y = 16 7x + y = 19

• (-2,5)

• (-2,-5)

• (2,-5)

• (2,5)