Q. Find the x-intercept of the line \(y=18x+6\).

Q: How do you find the x-intercept of the line y=18x+6?

Set y=0 and solve: 0=18x+6 gives x=-\tfrac{6}{18}=-\tfrac{1}{3}. The x-intercept is \left(-\tfrac{1}{3},0\right).

Q: What is the y-intercept of y=18x+6?.

The y-intercept is the value when x=0. Plugging in gives y=6, so the y-intercept is \left(0,6\right).

Q: What is the slope and what does it tell me?

The slope is 18 (from y=mx+b with m=18). It means for each increase of 1 in x, y increases by 18; the line is very steep upward.

Q: How can I graph this line quickly?

Plot the y-intercept \left(0,6\right), then use the slope rise/run = 18/1 to go up 18 units and right 1 unit (or reduce to up 2 right 1/9), or use another point like the x-intercept \left(-\tfrac{1}{3},0\right) and draw the line through them.

Q: Is the x-intercept the same as the root or zero of the function?

Is the x-intercept the same as the root or zero of the function?

Q: How do I check my answer algebraically?

Substitute x=-\tfrac{1}{3} into y=18x+6: y=18(-\tfrac{1}{3})+6=-6+6=0. Since y=0, the intercept is correct.

Q: What if the slope were zero or the line were vertical?

If slope m=0 (horizontal), then either no x-intercept (if y\neq0) or every x (if y=0). For a vertical line x=c, the x-intercept is (c,0) only if the line crosses the x-axis, i.e., if 0 is on the line.

Answer

Set y to 0: \(0=18x+6\). Solve: \(18x=-6\), \(x=-\frac{6}{18}=-\frac{1}{3}\).
x-intercept: \(\left(-\frac{1}{3},0\right)\)

Detailed Explanation

Find the x-intercept of the line

We are given the equation of the line

\( y = 18x + 6 \)

Understand what the x-intercept means.The x-intercept is the point where the graph crosses the x-axis. At that point, the y-coordinate equals 0. Therefore to find the x-intercept, set y equal to 0 in the equation and solve for x.
Set y to 0 and write the equation to solve.\( 0 = 18x + 6 \)
Isolate the term containing x.Subtract 6 from both sides to move the constant term:
\( 0 – 6 = 18x + 6 – 6 \)

which simplifies to

\( -6 = 18x \)
Solve for x by dividing both sides by 18.\( x = \dfrac{-6}{18} \)
Simplify the fraction.Both numerator and denominator have a common factor of 6, so divide top and bottom by 6:
\( x = \dfrac{-6 \div 6}{18 \div 6} = \dfrac{-1}{3} \)
State the x-intercept as a point and verify.The x-intercept is the point \( \left(-\tfrac{1}{3},\,0\right) \).
Verification: substitute \( x = -\tfrac{1}{3} \) into the original equation:

\( y = 18\left(-\tfrac{1}{3}\right) + 6 = -6 + 6 = 0 \), which confirms the result.

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Algebra FAQs

How do you find the x-intercept of the line \(y=18x+6\)?

Set \(y=0\) and solve: \(0=18x+6\) gives \(x=-\tfrac{6}{18}=-\tfrac{1}{3}\). The x-intercept is \(\left(-\tfrac{1}{3},0\right)\).

What does "x-intercept" mean?

The x-intercept is the point where the graph crosses the x-axis, so the y-coordinate is zero. Solve the equation with \(y=0\) to find the x-value.

What is the y-intercept of \(y=18x+6\)?.

The y-intercept is the value when \(x=0\). Plugging in gives \(y=6\), so the y-intercept is \( \left(0,6\right)\).

What is the slope and what does it tell me?

The slope is 18 (from \(y=mx+b\) with \(m=18\)). It means for each increase of 1 in \(x\), \(y\) increases by 18; the line is very steep upward.

How can I graph this line quickly?

Plot the y-intercept \(\left(0,6\right)\), then use the slope rise/run = 18/1 to go up 18 units and right 1 unit (or reduce to up 2 right \(1/9\)), or use another point like the x-intercept \(\left(-\tfrac{1}{3},0\right)\) and draw the line through them.

Is the \(x\)-intercept the same as the root or zero of the function?

How do I check my answer algebraically?

Substitute \(x=-\tfrac{1}{3}\) into \(y=18x+6\): \(y=18(-\tfrac{1}{3})+6=-6+6=0\). Since \(y=0\), the intercept is correct.

What if the slope were zero or the line were vertical?

If slope \(m=0\) (horizontal), then either no x-intercept (if \(y\neq0\)) or every x (if \(y=0\)). For a vertical line \(x=c\), the x-intercept is \((c,0)\) only if the line crosses the x-axis, i.e., if \(0\) is on the line.

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