What is the answer to this algebra problem?

Question

Write the standard form of the equation of the line described.

 

Parallel to -x+5y = 10 and goes through the point (-7,2)

Answer 1

Re-arranging the line -x + 5y = 10 into slope-intercept form gets you:

y = (1/5)x + 2

Parallel lines have equal slopes, so we know the slope of the second line is 1/5.

Use point-slope form to find the equation of the second line because we know its slope and a point that it goes through, (-7, 2):

y - y₁ = m(x - x₁)

y - 2 = (1/5) * (x + 7)

y = (1/5)x + 17/5

To re-arrange this into standard form, Ax + By = C, we move the x and y terms to the left-hand side:

(-1/5)x + y = 17/5

But remember, standard form requires A to be a positive integer. We multiply each term by -5 to get the desired result:

x - 5y = -17

Hope this was helpful. Credits to James S. for pointing out that A should be a positive integer.

Answer 2

To find a line parallel to a given line, you need the slope and a point on the line.

First let's find out what the slope is. When a line is parallel to another line, they have equal slopes. The slope of a line is easily recognized when it is in slope-intercept format.

-x + 5y = 10

Add x to both sides:

5y = x + 10

Divide both sides by the coefficient of y:

y = (1/5)x + 2

Now the line is in slope-intercept format, and the coefficient of x is the slope if this line.

We can either use point-slope form to find the y-intercept or the slope-intercept format, using the given point, to find the y-intercept. Then we will need to convert to standard format.

The slope-intercept method:

y - y₁ = m ( x -x₁ )

Plug in the values for the slope and the given point:

y - ( 2 ) = (1/5) * ( x - (-7) )

y -2 = ( 1/ 5 ) * ( x +7)

y = (1/5) x + 1.4 + 2

y = (1/5) x + 3.4

Now let's try the slope-intercept method:

y = mx + b

( 2) = (1/5) * (-7) + b

2 = -1.4 + b

3.4 = b

So y = (1/5) x +3.4, and we got the same result from both methods.

But we are not done yet. We still need to convert to standard form.

Multiply both sides by the denominator of the slope:

5y = x + ( 3.4 ) * 5

5y = x + ( 17 / 5 ) * 5

5y = x + 17

Subtract 5y from both sides, and subtract 17 from both sides:

-17 = x - 5y

If a = b, then b = a (the symmetric property of equality):

x - 5y = -17

The coefficients of x and y are integers, the coefficient of x ≥ 0, and there are no common factors of the coefficients of x and y and the constant term, other than 1.

So that is the standard form of the equation which is parallel to the original line and goes through the given point.