Write the equation in slope-intercept form:

\(\displaystyle y=mx+b\)

We were given the \(\displaystyle y\)-intercept, \(\displaystyle -1\), which means \(\displaystyle b=-1\):

\(\displaystyle y=mx-1\)

Given the \(\displaystyle x\)-intercept is \(\displaystyle 5\), the point existing on the line is \(\displaystyle (5,0)\). Substitute this point into the slope-intercept equation and then solve for \(\displaystyle m\) to find the slope:

\(\displaystyle 0=m(5)-1\)

Add \(\displaystyle 1\) to each side of the equation:

\(\displaystyle 1=5m\)

Divide each side of the equation by \(\displaystyle 5\):

\(\displaystyle m=\frac{1}{5}\)

Substituting the value of \(\displaystyle m\) back into the slope-intercept equation, we get:

 \(\displaystyle y=\frac{1}{5}x-1\)

By subtracting \(\displaystyle -\frac{1}{5}x\) on both sides, we can rearrange the equation to put it into standard form:

\(\displaystyle -\frac{1}{5}x+y=-1\)

To find the x-intercept, we need to find the value of \(\displaystyle x\) when \(\displaystyle y=0\).

So we first set \(\displaystyle y\) to zero.

\(\displaystyle y=4x+2\)

turns into

\(\displaystyle 0=4x+2\)

Lets subtract \(\displaystyle 4x\) from both sides to move \(\displaystyle x\) to one side of the equation.

\(\displaystyle -4x=4x-4x+2\)

After doing the arithmetic, we have

\(\displaystyle -4x=2\).

Divide by \(\displaystyle -4\) from both sides

\(\displaystyle \frac{-4x}{-4}=\frac{2}{-4}\)

\(\displaystyle x=-0.5\)

To find the y-intercept, we set \(\displaystyle x=0\)

So

\(\displaystyle y=x^2+x+9\)

turns into

\(\displaystyle y=0^2+0+9\).

After doing the arithmetic we get

\(\displaystyle y=9\).

The x-intercept can be found where \(\displaystyle y=0\)

So

\(\displaystyle y=3x+3\)

turns into

\(\displaystyle 0=3x+3\).

Lets subtract \(\displaystyle 3x\) from both sides to solve for \(\displaystyle x\).

\(\displaystyle -3x=3x-3x+3\)

After doing the arithmetic we have

\(\displaystyle -3x=3\).

Divide both sides by \(\displaystyle -3\)

\(\displaystyle \frac{-3x}{-3}=\frac{3}{-3}\)

\(\displaystyle x=-1\)

Rewrite the intercepts in terms of points.

X-intercept of 1: \(\displaystyle (1,0)\).

Y-intercept of 2: \(\displaystyle (0,2)\)

Write the slope-intercept form for linear equations.

\(\displaystyle y=mx+b\)

Substititute the y-intercept into the slope-intercept equation.

\(\displaystyle 2=m(0)+b\)

\(\displaystyle b=2\)

Substitute both the x-intercept point and the y-intercept into the equation to solve for slope.

\(\displaystyle 0=m(1)+2\)

\(\displaystyle -2=m\)

Rewrite by substituting the values of \(\displaystyle m\) and \(\displaystyle b\) into the y-intercept form.

\(\displaystyle y=-2x+2\)

In order for the equation to have x-intercepts at -1 and 6, it must have \(\displaystyle \small (x+1)\) and \(\displaystyle (x-6)\) as factors. This leaves us with only 2 choices,\(\displaystyle \small \small y = (x+1)(x-6)\) or \(\displaystyle \small y = \frac{-1}{3}(x+1)(x-6)\)

This equation must also have a y-intercept of 2. This means that plugging in 0 for x will gives us a y-value of 2. Because we have two options, we could plug in 0 for x in each to see which gives us an answer of 2:

a) \(\displaystyle \small y = (0+1)(0-6) = 1*-6 = -6\) we can eliminate that choice

b) \(\displaystyle \small y= -\frac{1}{3}(0+1)(0-6) = -\frac{1}{3}*1*-6 = 2\) this must be the right choice.

If we hadn't been given multiple options, we could have set up the following equation to figure out the third factor:

\(\displaystyle \small 2 = C(0+1)(0-6)\)

\(\displaystyle \small 2 = C*1*-6\)

\(\displaystyle \small 2 = -6C\) divide by -6

\(\displaystyle \small -\frac{1}{3} = C\)

We're writing the equation for a line passing through the points \(\displaystyle \small (0,-3)\) and \(\displaystyle \small (2,0)\). Since we already know the y-intercept, we can figure out the slope of this line and then write a slope-intercept equation.

To determine the slope, divide the change in y by the change in x:

\(\displaystyle \small \frac{-3 - 0 }{ 0 - 2 } = \frac{-3}{-2} = \frac{3}2\)

The equation for this line would be \(\displaystyle \small y = \frac{3}{2}x - 3\).

The line will eventually be in the form \(\displaystyle y = mx+b\) where \(\displaystyle b\) is the y-intercept.

The y-intercept in this case is \(\displaystyle 4\).

To find the equation, plug in \(\displaystyle 4\) for \(\displaystyle b\), and the other point, \(\displaystyle (-1,0)\) as x and y:

\(\displaystyle 0 = m(-1)+ 4\)

\(\displaystyle 0 = -m + 4\) add \(\displaystyle m\) to both sides

\(\displaystyle m = 4\)

This means the equation is \(\displaystyle y = 4x + 4\)