Grade 8 Module 4 Lesson 27

How many solutions will the following system of equations have?

 $y=\frac{1}{2}x+\frac{5}{2}$  
 $x-2y=7$  

• One solution

• No solutions

• Infinitely many solutions

Answer explanation

Get both equations into slope intercept form.
 $y=\frac{1}{2}x+\frac{5}{2}$  
 $y=\frac{1}{2}x-\frac{7}{2}$  
Since these two equations have the same slope of  $m=\frac{1}{2}$  , they are parallel lines.  Parallel lines never cross so there are no solutions.

How many solutions will the following system of equations have?

 $y=3x-2$  
 $-3x+y=-2$  

• One solution

• No solutions

• Infinitely many solutions

Answer explanation

First get both equations into slope intercept form.
 $y=3x-2$  
 $y=3x-2$  
Since the equations are the same, there are infinitely many solutions.

How many solutions will the following system of equations have?
 $y=\frac{3}{7}x-8$  

 $3x-7y=1$  

• One solution

• No solutions

• Infinitely many solutions

Answer explanation

First get both equations into slope intercept form.
 $y=\frac{3}{7}x-8$  
 $y=\frac{3}{7}x-\frac{1}{7}$  
Since these equations have the same slope of  $m=\frac{3}{7}$  , they are parallel lines.  Parallel lines never cross so there are no solutions.

How many solutions will the following system of equations have?

 $7x-10=y$  
 $y=5x+12$  

• One solution

• No solutions

• Infinitely many solutions

Answer explanation

Get both equations into slope intercept form.
 $y=7x-10$  
 $y=5x+12$  
Since these two equations have different slopes, there will be one solution.

How many solutions will the following system of equations have?

 $5y=\frac{15}{4}x+25$  
 $y=\frac{3}{4}x+5$  

• One solution

• No solutions

• Infinitely many solutions

Answer explanation

First get both equations into slope intercept form. 
 $y=\frac{3}{4}x+5$  
 $y=\frac{3}{4}x+5$  
Since these equations are the same, there will be infinitely many solutions

How many solutions will the following system of equations have?

 $6x-7y=\frac{1}{2}$  
 $12x-14y=1$  

• One solution

• No solutions

• Infinitely many solutions

Answer explanation

First get both equations into slope intercept form. 
 $y=\frac{6}{7}x-\frac{1}{14}$  
 $y=\frac{6}{7}x-\frac{1}{14}$  
Since these equations are the same, there will be infinitely many solutions

How many solutions will the following system of equations have?

 $x+9=y$  
 $x=4y-9$  

• One solution

• No solutions

• Infinitely many solutions

Answer explanation

Get both equations into slope intercept form.
 $y=x+9$  
 $y=\frac{1}{4}x+\frac{3}{2}$  
Since these two equations have different slopes, there will be one solution.

How many solutions will the following system of equations have?
 $5x-2y=6$  

 $-10x+4y=-14$  

• One solution

• No solutions

• Infinitely many solutions

Answer explanation

First get both equations into slope intercept form.
 $y=\frac{5}{2}x-3$  
 $y=\frac{5}{2}x-\frac{7}{2}$  
Since these equations have the same slope of  $m=\frac{5}{2}$  , they are parallel lines.  Parallel lines never cross so there are no solutions.

Use substitution to find the solution to the following system of equations.

 $x+9=y$  
 $x=4y-6$  

Your answer should be in the form (x,y).

Answer explanation

Equation 1:    $x+9=y$  
Equation 2:     $x=4y-6$  

Since equation 2 is solved for x, we will substitute equation 2 into equation 1.

 $\left(4y-6\right)+9=y$  
 $4y+3=y$  
 $3=-3y$  
 $-1=y$  

Now that we know  $y=-1$  , we can solve for x by plugging  $y=-1$  into either equation.

 $x=4\left(-1\right)-6$  
 $x=-4-6$  
 $x=-10$  

Finally, write your answer as an ordered pair. (-10,-1)

Use substitution to find the solution to the following system of equations.

 $7x-10=y$  
 $y=5x+12$  

Your answer should be in the form (x,y). 
(The number may be big.  That's ok!)

Answer explanation

Equation 1:    $7x-10=y$  
Equation 2:     $y=5x+12$  

Since equation 2 is solved for y, we will substitute equation 2 into equation 1.

 $7x-10=\left(5x+12\right)$  
 $2x-10=12$  
 $2x=22$  
 $x=11$  

Now that we know  $x=11$  , we can solve for y by plugging  $x=11$  into either equation.

 $y=5\left(11\right)+12$  
 $y=55+12$  
 $y=67$  

Finally, write your answer as an ordered pair. (11,67)