How do i solve this?

Question

f(x)=9x+7 and g(x)= x-8

Answer 1

When you are solving a system of equations - a system is a set of two or more - you are essentially looking for the point of intersection. This intersection also means the point that satisfies both equations. There are three algebraic methods you can use to solve: graphing, substitution, or elimination. I will explain all three below. Pick the method that works best for you.

f(x) = 9x + 7

g(x) = x - 8

I) Graphing: Graph each equation accurately on the same set of axes. It is important that the graphs are accurate because you are looking for the exact point of intersection. Both f(x) and g(x) are lines because they are in the form f(x) = mx + b. To graph f(x) = 9x + 7, place the y-intercept on (0, 7) and find the next point using the slope. A slope of 9 is the same as 9/1. The numerator has you move up (+) or down (-). The denominator has you move left (-) or right (+). In this case, both 9 and 1 are positive, so you would move up 9 right 1. To graph g(x) = x - 8, place the y-intercept on (0, -8) and move up 1 right 1 for a slope of 1 or 1/1. You can also use an online graphing tool like Desmos to graph for you. In this case, the online graphing tool is more helpful because they do not intersect at integer coordinates. Solution: (-1.875, -9.875)

II) Substitution: To use substitution, you need to make sure one variable is completely alone on one side. You would then plug in in for that variable into the other equation. I will change function notation to y in this case.

y = 9x + 7

y = x - 8

Since both y's are alone on one side, you can set both equations equal to each other because they are both equal to y.

9x + 7 = x - 8

Use algebra to solve for x. I will add 8 to both sides, then subtract 9x from both sides.

9x + 7 + 8 = x - 8 + 8

9x + 15 = x

9x - 9x + 15 = x - 9x

15 = -8x

Divide both sides by -8 to get x alone.

x = -1.875

To find y, plug back into either original equation. I will choose the second equation.

y = -1.875 - 8

y = -9.875

Solution: (-1.875, -9.875)

III) Elimination: To use elimination, pick one variable to have the same or opposite sign coefficient. You can choose both x's to have the same or opposite coefficient, or both y's to have the same or opposite coefficient. It is also important than each term is lined up. This means the y's are on top of each other, as well as the x's and the constants. In this case, both y's have the same exact coefficient of 1. I will choose to eliminate the y's. To eliminate y from y, you would subtract. The first step would be to subtract both equations.

y = 9x + 7

-

y = x - 8

__________

0 = 8x + 15

Be careful that when you subtract -8 from 7, you get a double negative that equates to positive 15. Now solve for x. Subtract 15 from both sides, then divide by 8.

-15 = 8x

x = -1.875

To find y, plug back in to either original equation. I will choose the first equation this time.

y = 9(-1.875) + 7

y = -9.875

Solution: (-1.875, -9.875)