Question

1. What is the solution to the system of equations?

Answer 1

Y=5x-7    -3x-2y=-12  (2,3)

y=4x+3   y=-x-2     (-1,-1)

12x+8y=140    5 adults 10 children
12(5)+8(10)=140
60+80=140

Answer 2

Answer: 1) A. (2, 3)

              2) B. (-1, -1)

              3) 10 children

Step-by-step explanation:

1) $\left \{ {{y=5x-7} \atop {-3x-2y=-12}} \right.$

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.