Question

Write an equation (a) in slope-intercept form and (b) in standard form for the line passing through (-1,6) and parallel to x + 2y = 7.

Answer 1

The equation in slope-intercept form for the line passing through (-1,6) and parallel to x + 2y = 7 is y = -1/2x + 5 1/2. The equation in standard form is -x + 2y = -14.

Step-by-step explanation:

Equation in Slope-Intercept Form (y = mx + b):

To find the equation of a line parallel to x + 2y = 7, we need to determine the slope and y-intercept of the given line. The slope-intercept form of an equation is y = mx + b, where m is the slope and b is the y-intercept.

Step 1: Rewrite the given equation in slope-intercept form:

x + 2y = 7

2y = -x + 7

y = -1/2x + 7/2

Step 2: Use the slope of the given line to determine the slope of the parallel line. Since the given line has a slope of -1/2, the parallel line will have the same slope.

Step 3: Use the point (-1, 6) and the slope (-1/2) to find the y-intercept of the parallel line using the formula y = mx + b:

6 = -1/2(-1) + b

6 = 1/2 + b

b = 5 1/2

Step 4: Substitute the slope (-1/2) and y-intercept (5 1/2) into the slope-intercept form equation to get the final equation:

y = -1/2x + 5 1/2

Equation in Standard Form (Ax + By = C):

To convert the equation to standard form, multiply through by 2 to eliminate the fraction:

2y = -x + 7

-x - 2y = -2x + 14

x + 2y = 2x - 14

x - 2x + 2y = -14

-x + 2y = -14

The equation in standard form for the line passing through (-1,6) and parallel to x + 2y = 7 is -x + 2y = -14.

Answer 2

the line equation we want is parallel to x+2y=7, which means they have the same slope. Rewritting this function in the slope form

`x+2y=7\Leftrightarrow y=-(x)/(2)+(7)/(2)`

The slope of our equation is (-1/2)!

Now, we just need to substitute the point to find the intercept.

`\begin{gathered} y=-(x)/(2)+b \\ 6=-((-1))/(2)+b\Rightarrow b=(11)/(2) \end{gathered}`

Now we have the slope and the intercept. Solving item (a) and writing this function in slope-intercept form gives us

`y=-(x)/(2)+(11)/(2)`

The standard form is:

`Ax+By=C`

Rewriting our function like this, we get:

`x+2y=11`

and this is the answer to item b.