MHB How do I find an equation of the line with a given x-intercept and point?

[mathdad]

Find an equation of the line that passes through (6, 2) and has the same x-intercept as the line -2x + y = 1.

As a first step, I must let y = 0 in the given equation.

-2x + y = 1

-2x + 0 = 1

-2x = 1

x = -1/2

The x-intercept is (-1/2, 0) creating the second point needed to find the slope.

What is next?

 

[MarkFL]

Okay, you now have two points on the line, so you can compute the slope, and then you have the slope and point(s) on the line, so the point-slope formula will be useful:

$$y-y_1=m\left(x-x_1\right)$$

:D

 

[mathdad]

Okay, you now have two points on the line, so you can compute the slope, and then you have the slope and point(s) on the line, so the point-slope formula will be useful:

$$y-y_1=m\left(x-x_1\right)$$

:D

Perfect. It's easier than I thought.

 

[HallsofIvy]

Equivalently, any (non-vertical) line can be written in the form y= ax+ b. Knowing that the line goes through (-1/2, 0) tells you that 0= a(-1/2)+ b. Knowing that the line goes through (6, 2) tells you that 2= a(6)+ b, giving two equations to solve for a and b. You can immediately eliminate b by subtracting one equation from the other which gives precisely the previous method.

 

[mathdad]

Equivalently, any (non-vertical) line can be written in the form y= ax+ b. Knowing that the line goes through (-1/2, 0) tells you that 0= a(-1/2)+ b. Knowing that the line goes through (6, 2) tells you that 2= a(6)+ b, giving two equations to solve for a and b. You can immediately eliminate b by subtracting one equation from the other which gives precisely the previous method.

Good data here.

 

[mathdad]

Find an equation of the line that passes through (6, 2) and has the same x-intercept as the line -2x + y = 1.

As a first step, I must let y = 0 in the given equation.

-2x + y = 1

-2x + 0 = 1

-2x = 1

x = -1/2

The x-intercept is (-1/2, 0) creating the second point needed to find the slope.

I will now solve the given equation for y.

-2x + y = 1

y= 2x + 1

The equation I need to find can be found by finding the slope using (-1/2, 0) and (6, 2).

I then plug one of the points and the slope into the point-slope formula. As a last step, solve the equation for y.