Question

Pilar says that two linear systems below have the same solution. (see picture) Is she correct? Explain.

Answer 1

The first set:

3x + 2y = 2  ---1)
5x + 4y = 6  ---2)

From 1), multiply all by 2, 6x + 4y = 4  ---3)

3) - 2),

6x + 4y - (5x + 4y) = 6 - 4
6x + 4y - 5x - 4y = 2
x = 2

Sub in x = 2 into 1),

3(2) + 2y = 2
2y = -4
y = -2

(2 , -2)


The second set:

3x + 2y = 2  ---1)
11x + 8y = 10  ---2)

From 1), multiply all by 4, 12x + 8y = 8 ---3)

3) - 2),

12x + 8y - (11x + 8y) = 8 - 10
12x + 8y - 11x - 8y = -2
x = -2

From this x value alone, we can tell that these two linear systems do NOT have the same solution as they meet at different coordinates.

Hope this helped! Ask me if there's any working from here that you don't understand! :)

Answer 2

Answer:

Pilar is correct because both system of equations are satisfied by the same solution.

Step-by-step explanation:

To know if both systems have the same solutions we need to solve one of them, and then evaluate that solution with the other system.

Let's solve the first system

$3x+2y=2\\5x+4y=6$

First, let's multiply the first equation by -2, then we sum both equations each other

$-6x-4y=-4\\5x+4y=6$

$-x=2\\x=-2$

Then, we use this value to find the other one

$3x+2y=2\\3(-2)+2y=2\\-6+2y=2\\2y=2+6\\y=\frac{8}{2}\\ y=4$

Therefore, the solution of the first system of equations is $(-2,4)$.

Now, let's evaluate this values in the second equation and see if they satisfy

$3x+2y=2\\3(-2)+2(4)=2\\-6+8=2\\2=2$

They satisfy the first equation, let's evaluate the second one

$11x+8y=10\\11(-2)+8(4)=10\\-22+32=10\\10=10$

They also satisfy the second equation. That means both system have the same solution.

Therefore, Pilar is correct because both system of equations are satisfied by the same solution.