Exercise 74 Page 173

The exercise tells us that Aimee thinks the solution to the system shown below is (-4,-6), and Eric thinks the solution is (8,2). We need to find who is correct and explain our reasoning. 2x- 3y = 10 6y=4x -20 Recall that for a point to be the solution of a system of equations it needs to satisfy both equations. Let's check whose solution gives a true statement when substituting it in both equations. We will check each of them individually.

Aimee's Solution

We will check Aimee's solution first. We will start by substituting (-4,-6) in the first equation.

2x- 3y = 10

SubstituteII

x= -4, y= -6

2( -4) -3( -6) ? = 10

Multiply

Multiply

-8 +18 ? = 10

AddTerms

Add terms

10 = 10

We can see that Aimee's solutions satisfy the first equation. Let's check if it also satisfies the second one.

6y=4x -20

SubstituteII

x= -4, y= -6

6( -6) ? =4( -4)-20

Multiply

Multiply

-36 ? = -16-20

SubTerm

Subtract term

-36 = -36

Once more we obtained a true statement. Aimee's solution satisfies both equations, and therefore her solution is correct.

Eric

We will start by substituting Eric's solution, (8,2), in the first equation.

2x- 3y = 10

SubstituteII

x= 8, y= 2

2( 8) -3( 2) ? = 10

Multiply

Multiply

16 -6 ? = 10

SubTerm

Subtract term

10 = 10

Eric's solutions satisfies the first equation. Let's now check to see if it also works for the second equation.

6y=4x -20

SubstituteII

x= 8, y= 2

6( 2) ? =4( 8)-20

Multiply

Multiply

12 ? = 32-20

SubTerm

Subtract term

12 = 12

As Eric's solution satisfies both equations, he is correct as well.

Conclusions

After checking both Aimee and Eric's solutions, we can see that both are correct. How can this be?

If we manipulate one of the equations from our system following the Properties of Equality, we can find our answer.

2x- 3y = 10 & (I) 6y=4x -20 & (II)

MultEqn

(I):LHS * 2=RHS* 2

2(2x- 3y = 10) 6y=4x -20

Distr

(I):Distribute 2

2*2x- 2*3y = 2*10 6y=4x -20

Multiply

(I):Multiply

4x- 6y = 20 6y=4x -20

AddEqn

(I):LHS+6y=RHS+6y

4x- 6y +6y = 20+ 6y 6y=4x -20

AddTerms

(I):Add terms

4x = 20+ 6y 6y=4x -20

SubEqn

(I): LHS-20=RHS-20

4x-20 = 20+ 6y -20 6y=4x -20

SubTerms

(I):Subtract terms

4x-20 = 6y 6y=4x -20

RearrangeEqn

(I):Rearrange equation

6y=4x -20 6y=4x -20

As we can see, both equations are equivalent. Therefore there will be infinitely many solutions for the system. Every point satisfying 6y=4x -20 will be a solution to the system.