Question 442·Medium·Algebra

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Line $k$ is defined by the equation $4y - 8x = 12$ . Line $m$ is parallel to line $k$ and passes through the point $(6,-5)$ .

Which of the following is an equation of line $m$ ?

An equation for one line, a parallel condition, and a point on another line give you what you need to write the new equation. Graph the given line in Desmos to check its direction, then solve for $y$ to read its exact slope. Keep that slope and use the point to find the new intercept. Don't copy the old intercept: parallel lines can cross the $y$ -axis at different places.

Hints

Hint 1
In slope-intercept form, $y=mx+b$ , the number multiplying $x$ is the slope. Get $y$ by itself in the equation for line $k$ before trying to read that number.
Hint 2
Parallel lines have the same slope: they rise or fall at the same rate. Once you know line $k$ 's slope, what must the number multiplying $x$ be in line $m$ 's equation?
Hint 3
A point $(x,y)$ on a line makes its equation true. Put the given point's coordinates into $y=mx+b$ to find the $y$ -intercept $b$ . The point's $y$ -coordinate isn't automatically $b$ unless its $x$ -coordinate is $0$ .

Step-by-step

Keep the slope, then use the point
Step 1
Check the given line's direction
Type $4y-8x=12$ in Desmos. It draws a rising line. Click where it crosses the $y$ -axis to see $(0,3)$ . That crossing belongs to line $k$ ; we'll find line $m$ 's own crossing later.
Step 2
Move the $x$ term
To read line $k$ 's slope, get $y$ by itself. Add $8x$ to both sides:
$4y=8x+12.$
Step 3
Read line $k$ 's slope
Divide every term by $4$ : $y=2x+3$ . In slope-intercept form, $y=mx+b$ , the number multiplying $x$ is the slope. So line $k$ 's slope is $2$ , consistent with its rising graph.
Step 4
Give line $m$ the same slope
Parallel lines have the same slope, so line $m$ must have slope $2$ . Write it as
$y=2x+b,$
where $b$ is its still-unknown $y$ -intercept, the value of $y$ when $x=0$ . Don't change the slope's sign.
Step 5
Use the point on line $m$
The point $(6,-5)$ says $x=6$ gives $y=-5$ on line $m$ . Substitute both coordinates into its equation: $-5=2(6)+b$ . The $-5$ isn't the intercept because the point's $x$ -coordinate isn't $0$ .
Step 6
Find the new intercept
Type $-5\sim2(6)+b$ in Desmos. The $\sim$ asks Desmos to find the unknown $b$ ; under PARAMETERS, it shows $b=-17$ .
Step 7
Write line $m$ 's equation
Put the slope $2$ and intercept $-17$ into $y=mx+b$ to get $y=2x-17$ . It has the same slope as line $k$ and passes through $(6,-5)$ . A parallel line keeps the slope; the given point determines its intercept. Choice D.

Question 442·Medium·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

Hints

Step-by-step

Lessons that teach this