Complete the system of equations with a linear equation so that the graphs of the equations intersect at their x- and y-intercepts.

Question

Complete the system of equations with a linear equation so that the graphs of the equations intersect at their x- and y-intercepts.

y=x^3+4x+3x+12

y=?

Answer 1

First, we need to find the intercepts.

For the x intercepts, yet y=0 and solve. (I believe there is a typo in the problem and the second term should be squared)

0=x³+4x²+3x+12

Factor by grouping:

0=x²(x+4)+3(x+4)

0=(x²+3)(x+4)

Set factors =0 and solve. The first factor doesn't give a solution.

x+4=0 Subtract 4 from both sides

x+4-4=0-4

x=-4

The x intercept is (-4,0)

For the y intercepts, yet x=0 and solve.

y=(0)³+4(0)²+3(0)+12

y=12

The y intercept is (0,12)

Next, find the equation of the line that contains these 2 points. First, find the slope.

m=(y2-y1)/(x2-x1)

m=(12-0)/(0-(-4)) = 12/3

m=3

b, we already have b=12

So the equation of the line is y=3x+12