The points (-2, -4), (0,0), (4,a) and (b, 18) is on a straight line . What is the number a and b?
I wonder if it is possible to calculate the answer to the question, or are you just simply doing guesswork?
I don't know if I'm solving it the right way.
The answer is a = 8 and b = 9
No..we don't usually like to do too much "guessing" on here....
Since all the points lie on a line, the slope between them will be the same. So the slope between (-2, -4) and (0, 0) = [-4 - 0]/ [-2, 0] = -4/-2 = 2
So, using this, the slope between (0, 0) and (4, a) = 2 ....and we have..
[a - 0] / [4 - 0] = 2 → a/4 = 2 → (multiply both sides by 2)......a = 8.
[18 - 0 ] / [ b - 0] = 2 → 18/2 = b → 9 = b
And that's it .....(without the guesswork .....!!!)
Points (-2, -4), (0,0), (4,a) and (b, 18)
If they're all on the same line, then they're all on the same slope, or share the same "m" value in y=mx+b. Now, in the first two ordered pairs, the x values are -2 and 0. The y values are -4 and 0. For each respective ordered pair, the equations can be made:
If you take the second equation, you can solve for b easily, because anything times 0 is 0.
So, since this is all on the same line, or slope, that value of b works in the other equation.
Now, solve for m! You can drop the plus 0, and then divide by -2 on both sides.
So, now we know m=2 and b=0. We can plug these into our slope-intercept equation. From now on:
Now, we need to solve for (4,a) and (b, 18).
In the first one, we know the x value, 4, but need to find y, which is a.
In the second one, we know the y value, 18, but need to find x, which is b.
So you were right! a=8 and b=9.
SIDE NOTE: Because one ordered pair was (0,0), we could skip a step to find b in y=mx+b. Normally, this is how it would be done:
Find slope-intercept form of (-2,-4) and (0,0)
Rewrite both equations so b is left alone. On the first one, subtract m(-2) from both sides.
This could be rewritten as -(-2m), or simply +2m.
Now, subtract m(0) from both sides of the second equation
So, if -4+2m=b, and b=0-m0, we can squish that all together:
Then, you would go about solving for m like you would in any equation by moving all the b values to one side and all the constants (numbers without variables) to the other. We will add 0 to both sides, and subtract -2m from both sides. This will put the constants on the left and the m values on the right.
Now, dividing -4/-2 gets:
Just like it did before! Now you plug this back into the equations (the ones using points (-2,-4) and (0,0) in the slope-intercept equation y=mx+b) to get your b value. Then solve the rest like you did already!
Thanks guys so much for the answers! People like you are gold when you're trying to learning math without a teacher!