Q. Drag the red and blue dots along the \(x\)-axis and \(y\)-axis to graph \(-3x + 6y = 21\).

Q: What are the x - and y -intercepts of -3x+6y=21?.

Set y=0: -3x=21 so x=-7. Set x=0: 6y=21 so y=21/6=7/2. Intercepts: (-7,0) and (0,7/2).

Q: How do I convert -3x+6y=21 to slope-intercept form?

Solve for y: 6y=3x+21, so y=(1/2)x+7/2. Slope m=1/2, y-intercept b=7/2..

Q: Where should I place the red and blue dots on the axes?

Place one dot at the x-intercept \left(-7,0\right) on the x-axis and the other at the y-intercept \left(0,\frac{7}{2}\right) on the y-axis.

Q: How do I draw the line after placing the dots?

After placing dots at (-7,0) and (0,7/2) , draw a straight line through both points; that line is the graph of -3x+6y=21.

Q: What if the grid ticks are integers and \frac{7}{2} is between marks?

Place the y-axis dot halfway between 3 and 4 at 3.5; precision matters but approximate mid-tick is fine. Use a ruler to keep the line straight.

Q: How is the slope interpreted on the graph?

Slope m=\frac{1}{2} means rise/run =1 up for every 2 right. From (0,\frac{7}{2}), go right 2 and up 1 to get another point on the line.

Q: Can I simplify the equation before graphing?

Yes, divide both sides by 3 to get -x+2y=7 , or rearrange to y=\left(\frac{1}{2}\right)x+\frac{7}{2} ; both forms make intercepts and slope easier to find.

Answer

Set \( y = 0 \):

\[ -3x + 6(0) = 21, \quad -3x = 21, \quad x = -7 \]

Detailed Explanation

Goal: Graph the line given by the equation
\[
-3x + 6y = 21.
\]
I will find the x-intercept and y-intercept (these are the points you will place the dots on) and then connect them to draw the line.
Find the x-intercept (set y equal to 0):Substitute y = 0 into the equation and solve for x.\[
-3x + 6(0) = 21
\]
\[
-3x = 21
\]
Divide both sides by -3:
\[
x = \frac{21}{-3} = -7.
\]

Therefore the x-intercept is the point \((-7,\,0)\).

Action: Drag the dot on the x-axis to the coordinate (-7, 0). (If you must choose a color, place the red dot at (-7, 0).)
Find the y-intercept (set x equal to 0):Substitute x = 0 into the equation and solve for y.\[
-3(0) + 6y = 21
\]
\[
6y = 21
\]
Divide both sides by 6:
\[
y = \frac{21}{6} = \frac{7}{2} = 3.5.
\]

Therefore the y-intercept is the point \((0,\,\tfrac{7}{2})\) or \((0,\,3.5)\).

Action: Drag the dot on the y-axis to the coordinate (0, 7/2). (If you must choose a color, place the blue dot at (0, 7/2).)
Draw the line through the two intercepts:Now plot the two points you placed: (-7, 0) on the x-axis and (0, 7/2) on the y-axis. Draw a straight line through these two points. That line is the graph of the equation.Optional verification by converting to slope-intercept form:
\[
-3x + 6y = 21
\]
Add 3x to both sides:
\[
6y = 3x + 21
\]
Divide by 6:
\[
y = \tfrac{1}{2}x + \tfrac{7}{2}.
\]

The y-intercept is \((0, \tfrac{7}{2})\), and the slope is \(\tfrac{1}{2}\). A line with slope \(\tfrac{1}{2}\) passing through \((0, \tfrac{7}{2})\) indeed passes through \((-7, 0)\). This confirms the two plotted intercepts are correct.

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Algebra FAQs

What are the x- and y-intercepts of \(-3x+6y=21\)?.

Set \(y=0\): \(-3x=21\) so \(x=-7\). Set \(x=0\): \(6y=21\) so \(y=21/6=7/2\). Intercepts: \((-7,0)\) and \((0,7/2)\).

How do I convert \(-3x+6y=21\) to slope-intercept form?

Solve for y: \(6y=3x+21\), so \(y=(1/2)x+7/2\). Slope \(m=1/2\), \(y\)-intercept \(b=7/2\)..

Where should I place the red and blue dots on the axes?

Place one dot at the \(x\)-intercept \(\left(-7,0\right)\) on the \(x\)-axis and the other at the \(y\)-intercept \(\left(0,\frac{7}{2}\right)\) on the \(y\)-axis.

How do I draw the line after placing the dots?

After placing dots at \( (-7,0) \) and \( (0,7/2) \), draw a straight line through both points; that line is the graph of \(-3x+6y=21\).

What if the grid ticks are integers and \( \frac{7}{2} \) is between marks?

Place the \(y\)-axis dot halfway between 3 and 4 at 3.5; precision matters but approximate mid-tick is fine. Use a ruler to keep the line straight.

How can I check my plotted line is correct?

How is the slope interpreted on the graph?

Slope \(m=\frac{1}{2}\) means rise/run \(=1\) up for every \(2\) right. From \((0,\frac{7}{2})\), go right \(2\) and up \(1\) to get another point on the line.

Can I simplify the equation before graphing?

Yes, divide both sides by 3 to get \( -x+2y=7 \), or rearrange to \( y=\left(\frac{1}{2}\right)x+\frac{7}{2} \); both forms make intercepts and slope easier to find.

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