Write the equation of a line passing through points (1,3) and (2,5) in point-slope, slope-intercept and standard form

Answer 1

slope = m = (5-3) / (2-1) = 2

y - 3 = 2(x - 1) this is point-slope form.

y = 2x - 2 + 3

y = 2x + 1 this is slope-intercept form.

2x - y = -1 this is standard form.

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Thank you!

Best of luck to you :)

John B.

Atlanta, GA

Answer 2

Hi Aisha (and anyone else working on a similar problem)!

Here's 3 math dominoes for you to knock over one at a time for this problem:

If you're given two points, then you can find the slope.

If you have a point and the slope, then you can use point-slope form.

If you have the equation of the line in point-slope form, then you can format it into the other two forms!

Let's knock these dominoes over, shall we? Take a look...

If you're given two points, then you can find the slope:

The formula for slope is m = (y₂-y₁)/(x₂-x₁).

A little trick: Label your points (x₁,y₁) and (x₂,y₂) to help you fill in the formula and find your slope with more ease!

(1,3) and (2,5)

(x₁,y₁) and (x₂,y₂)

m = (y₂-y₁)/(x₂-x₁)

m = (5-3)/(2-1)

m = 2/1

or m = 2

Ok, now if you have a point and the slope, then you can use point-slope form:

Pick a point, any point. I'm choosing (1,3), but the point (2,5) works, too!

Point: (1,3) or (x₁,y₁)

Slope: m = 2

Point slope form: y-y₁ = m(x-x₁)

Fill in the variables that you have (x₁, y₁ and m):

y-y₁ = m(x-x₁)

y-3 = 2(x-1) --> This is technically point-slope form. Boom, done!

Now, if you have the equation of the line in point-slope form (like we do), then you can format it into the other two forms, slope-intercept form + standard form:

Slope-intercept form (hint, we just need to isolate y from the point-slope form we just found!):

y = mx + b

Let's go from y-3 = 2(x-1) --> y = mx + b

y-3 = 2(x-1)

y-3 = 2x-2 *Distribute (or multiply) the 2

+3 +3 *Add 3 to both sides and boom! You've got slope-intercept form

y = 2x+1

y = mx+b *Woohoo! (Verify that m=2 just by looking at it; nice work.)

Now for standard form:

ax + by = c (We just want the x term and the y term on the left side of the equal sign and the regular ol' integer on the other, that's all. Easy-peasy.)

y = 2x+1

-2x -2x *Move it on over

-2x + y = 1

***One last thing, and it's important: In standard form, we want the x term to be positive; x>0)

So, take -2x + y = 1 *and multiply everything by -1, that'll take care of the x and consequently change the sign of all the other terms:

2x - y = -1

ax + by = c *And there we have it, standard form.

So, to recap, if all you have is two points, you can write the equation of the line that passes through those points in three different forms:

Point-slope: y-3 = 2(x-1) --- y-y₁ = m(x-x₁)

Slope-intercept: y = 2x+1 --- y = mx+b

Standard: 2x - y = -1 --- ax + by = c

BONUS: Do you know what the intercept or b is all about from the slope-intercept form? It's the y-intercept, or the point where the line crosses exactly over the y-axis. I like to say, it's where the y-team intercepts the ball. So, in this case, it's b=1 or the y-intercept is (0,1).

I hope this helps y'all! Happy mathing!