What is the solution to the system of equations?

y = 5x - 7

-3x - 2y = -12

A(2, 3)
B(1, -2)
C(-2, 9)
D(3, 8)

What is the solution to the systems of equations represented by the 2 equations?

y = 4x + 3

y = -x - 2

A(1, 7)
B(-1, -1)
C(2, -4)
D(-3, -9)

A family went to an amusement park and paid $12 for each adult and $8 for each child. A group of 15 people went to the park and it cost $140. This system of equations models this situation, where x is the number of adults and y is the number of children.

How many children were in the group?

x + y = 15

12x + 8y = 140

Since the y is already isolated in the first equation, to solve the system you simply substitute that expression into the second equation and then solve, finding the value of x:

$-3x-2(5x-7)=-12\\-3x-10x+14=-12\\-13x=-12-14\\-13x=-26\\x=\frac{26}{13}=2$

And then you substitute that value into the first equation and solve to find the value of y:

$y=5(2)-7\\y=10-7\\y=3$

So, the solution of the system is (2, 3).

2) $\left \{ {{y=4x+3} \atop {y=-x-2}} \right.$

Since the y is already isolated in both equations, to solve the system you simply equalize the first and the second expression and then solve, finding the value of x:

$4x+3=-x-2\\4x+x=-2-3\\5x=-5\\x=-1$

And then you substitute that value into the first or the second equation (whichever you like) and solve to find the value of y:

$y=-(-1)-2\\y=1-2\\y=-1$

So, the solution of the system is (-1, -1).

3) $\left \{ {{x+y=15} \atop {12x+8y=140}} \right.$

To solve the system, the easiest way is to isolate the y in the first equation and then substitute the expression obtained into the second equation, finding the value of x.

From the first equation: $y=15-x$

Substituting:

$12x+8(15-x)=140\\12x+120-8x=140\\4x=140-120\\4x=20\\x=5$

And then you substitute that value into the first equation and find the value of y:

$y=15-5\\y=10$

So, there were 10 children in the group.