Q. Find the x-intercept of the line \(3x + 6y = 21\).

Q: How do you find the x-intercept of a line?

Set y to 0 and solve. For example, for 3x+6y=21 set y=0 giving 3x=21, so x=7. The x-intercept is (7,0).

Q: What is the y-intercept of 3x+6y=21?

Set x=0: 6y = 21 so y = \frac{7}{2}. The y-intercept is \left(0, \frac{7}{2}\right).

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

Solve for y: 6y = 21 - 3x so y = -\frac{1}{2}x + \frac{7}{2}. That is the slope-intercept form.

Q: What is the slope of the line?

From y = -\frac{1}{2}x + \frac{7}{2} the slope is -\frac{1}{2}.

Q: How can I graph this line quickly?

Plot the intercepts (7,0) and (0, \frac{7}{2}), then draw the straight line through them.

Q: Are the intercepts integers?

The x-intercept 7 is an integer; the y-intercept \frac{7}{2}=3.5 is not.

Q: Does this line pass through the origin?

No. Plugging (0,0) gives 0\neq21, so the origin is not on the line.

Answer

Set \(y=0\). Then \(3x=21\), so \(x=7\).

X-intercept: \((7,0)\)

Detailed Explanation

Solution — step-by-step

Definition: The x-intercept is the point where the line crosses the x-axis, so the y-coordinate is 0. Therefore set y = 0.
Substitute y = 0 into the equation 3x + 6y = 21:

\[3x + 6(0) = 21\]
Simplify the left side:

\[3x + 0 = 21\]

\[3x = 21\]
Isolate x by dividing both sides by 3:

\[x = \frac{21}{3}\]

\[x = 7\]
Thus the x-intercept is the point with coordinates

\[(7,0)\]
Check by substitution back into the original equation:

\[3(7) + 6(0) = 21\]

\[21 + 0 = 21\]

Equality holds, so the result is correct.

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FAQs

How do you find the x-intercept of a line?

Set y to 0 and solve. For example, for \(3x+6y=21\) set \(y=0\) giving \(3x=21\), so \(x=7\). The x-intercept is \((7,0)\).

What is the y-intercept of \(3x+6y=21\)?

Set \(x=0\): \(6y = 21\) so \(y = \frac{7}{2}\). The y-intercept is \(\left(0, \frac{7}{2}\right)\).

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

Solve for \(y\): \(6y = 21 - 3x\) so \(y = -\frac{1}{2}x + \frac{7}{2}\). That is the slope-intercept form.

What is the slope of the line?

From \(y = -\frac{1}{2}x + \frac{7}{2}\) the slope is \(-\frac{1}{2}\).

How can I graph this line quickly?

Plot the intercepts \((7,0)\) and \((0, \frac{7}{2})\), then draw the straight line through them.

Can I write the equation in intercept form?

Are the intercepts integers?

The x-intercept 7 is an integer; the y-intercept \(\frac{7}{2}=3.5\) is not.

Does this line pass through the origin?

No. Plugging \((0,0)\) gives \(0\neq21\), so the origin is not on the line.

Find the x-intercept by setting y=0.
Solve for x.

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