Q. Find the x-intercept of the line \(7x+12y=-14\).

Q: What is an x-intercept?

The x-intercept is where a graph crosses the x-axis, so y = 0. Its coordinate is (x, 0).

Q: How do I find the x-intercept of 7x+12y=-14?.

Set y=0: 7x+12(0)=-14 so 7x=-14, x=-2. The x-intercept is (-2,0).

Q: How do you find an x-intercept from a general line ax+by=c?.

Set y=0. Then ax=c so x=\dfrac{c}{a}, provided a\neq 0. The intercept is \bigl(\dfrac{c}{a},0\bigr).

Q: How do I find the y-intercept of this line?

Set x=0. 12y=-14 so y=-\dfrac{7}{6}. The y-intercept is \left(0,-\dfrac{7}{6}\right).

Q: How can I graph the line using intercepts?

Plot the x-intercept (-2,0) and y-intercept \left(0,-\dfrac{7}{6}\right), then draw the straight line through them.

Q: What if the x-coefficient is zero (a=0)?

What if the x-coefficient is zero (a=0)?

Q: How do I write the line in intercept form x/a+y/b=1?.

The x-intercept a=-2 and y-intercept b=-\dfrac{7}{6}. So \dfrac{x}{-2}+\dfrac{y}{-\dfrac{7}{6}}=1, which simplifies appropriately.

Q: How is slope related to the intercepts?

Slope m equals rise/run between intercepts: m=\dfrac{y_2-y_1}{x_2-x_1}. For (-2,0) and \left(0,-\dfrac{7}{6}\right), m=\dfrac{-\dfrac{7}{6}-0}{0-(-2)}=-\dfrac{7}{12}.

Answer

Set y = 0: \(7x + 12(0) = -14\).
So \(7x = -14\) and \(x = -2\).
x-intercept: \((-2, 0)\)

Detailed Explanation

Solution

Definition: The x-intercept is the point where the line crosses the x-axis. Every point on the x-axis has y equal to 0, so to find the x-intercept set y equal to 0.
Substitute y = 0 into the equation 7x + 12y = −14:
\[7x + 12(0) = -14\]
Simplify the left side, since 12 times 0 is 0:
\[7x + 0 = -14\]

\[7x = -14\]
Solve for x by dividing both sides of the equation by 7. Division is valid because 7 is not zero:
\[x = \frac{-14}{7} = -2\]
The x-intercept is the point with x-coordinate −2 and y-coordinate 0. Therefore the x-intercept is
\[(-2,\,0)\]

You can verify by substituting back into the original equation:

\[7(-2) + 12(0) = -14 + 0 = -14,\]

which matches the right-hand side, confirming the result.

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Algebra FAQs

What is an \(x\)-intercept?

The x-intercept is where a graph crosses the x-axis, so \(y = 0\). Its coordinate is \((x, 0)\).

How do I find the x-intercept of \(7x+12y=-14\)?.

Set \(y=0\): \(7x+12(0)=-14\) so \(7x=-14\), \(x=-2\). The x-intercept is \((-2,0)\).

How do you find an x-intercept from a general line \(ax+by=c\)?.

Set \(y=0\). Then \(ax=c\) so \(x=\dfrac{c}{a}\), provided \(a\neq 0\). The intercept is \(\bigl(\dfrac{c}{a},0\bigr)\).

How do I find the \(y\)-intercept of this line?

Set \(x=0\). \(12y=-14\) so \(y=-\dfrac{7}{6}\). The y-intercept is \(\left(0,-\dfrac{7}{6}\right)\).

How can I graph the line using intercepts?

Plot the x-intercept \((-2,0)\) and y-intercept \(\left(0,-\dfrac{7}{6}\right)\), then draw the straight line through them.

What if the \(x\)-coefficient is zero (\(a=0\))?

How do I write the line in intercept form \(x/a+y/b=1\)?.

The x-intercept \(a=-2\) and y-intercept \(b=-\dfrac{7}{6}\). So \( \dfrac{x}{-2}+\dfrac{y}{-\dfrac{7}{6}}=1\), which simplifies appropriately.

How is slope related to the intercepts?

Slope \(m\) equals rise/run between intercepts: \(m=\dfrac{y_2-y_1}{x_2-x_1}\). For \((-2,0)\) and \(\left(0,-\dfrac{7}{6}\right)\), \(m=\dfrac{-\dfrac{7}{6}-0}{0-(-2)}=-\dfrac{7}{12}\).

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