Question

Can someone help me with this algebra two assignment I will give brainilest

Answer 1

The answer is incorrect because the slope formula was used incorrectly in step 1 (parentheses weren't used during the substitution and/or the wrong values were substituted).

y=4x-6

Explanation:

The equation we're looking for in slope intercept form looks like
`y=mx+b`, where "m" is the slope, "b" is the y-coordinate of the y-intercept, and "x" and "y" are variables that have a certain mathematical relationship through this equation.

To find the equation itself, it usually is easier to find slope before finding the y-intercept.

Finding the slope

There are many ways to represent slope (usually represented by the letter m), and while they all mean the same thing, they look different, some examples are shown below:

`m=(rise)/(run) =\frac{\text{horizontal change}}{\text{vertical change}}=\frac{\Delta{y}}{\Delta{x}} =\frac{\text{change in }y}{\text{change in }x}=(y_2-y_1)/(x_2-x_1)`

For this problem, since we are given ordered pairs, probably the most useful version will be the last one:

`m=(y_2-y_1)/(x_2-x_1)`

Substituting the values from the ordered pairs, it's important to remember that the ordered pairs are (x,y)... that the first coordinate is an x (and will go into the denominator, or bottom, of the fraction), and the second coordinate is a y (and will go into the numerator, or top of the fraction).

It actually doesn't matter which point we choose to be point 1, and which point we choose to be point 2. For the least amount of confusion, I choose the first point they gave to be point 1, and the second point they gave to be point 2.

So,
`\text{Point 1: } (1,-2)`, and
`\text{Point 2: } (3,6)`

If I could give only one piece of advice, it would be "ANY time you substitute, use parentheses." You can always simplify later, but for the initial substitution, keep them (this is why the "mistake" happened in this problem, and is the thing we need to fix).

`m=(y_2-y_1)/(x_2-x_1)`

`m=((6)-(-2))/((3)-(1))`

Simplifying...

`m=((6)+(2))/((3)-(1))`

`m=(8)/(2)`

`m=4`

Thus, the equation we want in slope intercept form, with the new information we know plugged in, looks like this:

`y=(4)x+b\\y=4x+b`

"b" is still the y-coordinate of the y-intercept (which we don't know yet), and "x" and "y" are variables related through this mathematical equation.

Finding the y-intercept

With "m" solved for, we can solve for the "b" by using the relationship in the equation we have so far.

We know that every "x" and "y" ordered pair is an "x" related to the "y" by this specific equation. Even though we don't know the specific value of "b", we know that the equation must be true for every ordered pair on the line.

While we don't know many associated "y" values and "x" values (for instance, we don't know what the "y" value is when the x is 10, and we don't know what the x value is the "y" is 26), we do know two associations for sure... from the two ordered pairs

`\text{Point 1: } (1,-2)`, and
`\text{Point 2: } (3,6)`, so

`x=1 \text{ is related to } y=-2`, and
`x=3 \text{ is related to } y=6`

... and they're related through this equation that we've almost found!

Now, in our equation, there is only room to associate one x and y value at a time, so we need to pick one known association, and substitute those values in. Since this equation is supposed to relate the x to the y, it should be true for both points (so, we can try both, and it should work out the same. If you don't want to try both, if you see one ordered pair that has numbers that look easier, use that one):

Substituting Point 1 to find b

`y=4x+b`

`(-2)=4(1)+b`

simplifying the right side, 4*1 is 4...

`(-2)=4+b`

subtracting 4 from both sides of the equation, and simplifying the left hand side

`-2-4=b\\-2+-4=b\\-6=b`

Substituting Point 2 to find b (not necessary, but a good double-check)

`y=4x+b`

`(6)=4(3)+b`

simplifying the right side, 4*3 is 12...

`6=12+b`

subtracting 4 from both sides of the equation, and simplifying the left hand side

`6=12+b\\6-12=b\\6+-12=b\\-6=b`

So, as a verification, we get the same value of b (which we should have, because this equation is supposed to work for every pair of (x,y) that has this relationship, and those were the only two points that we knew that did).

The Final equation in slope intercept form

Substituting the value we found for b:

`y=4x+(-6)`

...and simplifying...

`y=4x-6`