Q. Find the x-intercept of the Line \( 6x – 4y = 14 \)

Answer

Set y to 0.
The x-intercept occurs where y = 0.

\[ 6x – 4(0) = 14 \]
Simplify the equation.
\[ 6x = 14 \]
Solve for x.
Divide by 6 and simplify.

\[ x = \frac{14}{6} = \frac{7}{3} \]
State the x-intercept.
\[ \left(\frac{7}{3}, 0\right) \]

Detailed Explanation

Solution

Definition: The x-intercept of a line is the point where the line crosses the x-axis, which occurs when the y-coordinate equals 0.

Set y equal to 0 in the given equation \(6x – 4y = 14\):

\(6x – 4(0) = 14\).

Explanation: Replacing y by 0 gives the equation that x must satisfy at the x-intercept.
Simplify the left side: \(6x – 0 = 14\), so

\(6x = 14\).

Explanation: Multiplying -4 by 0 yields 0, leaving \(6x\) on the left.
Solve for x by dividing both sides by 6:

\(x = \dfrac{14}{6}\).

Explanation: Dividing isolates x; simplify the fraction by dividing numerator and denominator by 2 to get

\(x = \dfrac{7}{3}\).
Write the x-intercept as an ordered pair:

\(\left(\dfrac{7}{3},\,0\right)\).

Explanation: The x-intercept has x-coordinate \(7/3\) and y-coordinate 0.
Optional check: Substitute \(x=\dfrac{7}{3}\) and \(y=0\) into the original equation:

\(6\left(\dfrac{7}{3}\right)-4(0)=6\cdot\dfrac{7}{3}=2\cdot 7=14\), which matches the right side.

Explanation: The point satisfies the equation, confirming the result.

Answer: \(\left(\dfrac{7}{3},\,0\right)\)

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Frequently Asked Questions

What is the x-intercept of 6x - 4y = 14?

Set y = 0: 6x = 14, so x = 14/6 = 7/3. The x-intercept is (7/3, 0).

How do I find the y-intercept?

Set x = 0: -4y = 14, so y = -14/4 = -7/2. The y-intercept is (0, -7/2).

How do I convert the equation to slope-intercept form?

Solve for y: -4y = 14 - 6x, so y = (6/4)x - 14/4, which simplifies to y = (3/2)x - 7/2.

What is the slope of the line?

From y = (3/2)x - 7/2, the slope is 3/2.

How can I graph the line quickly?

Plot the x-intercept (7/3, 0) and y-intercept (0, -7/2), then draw the straight line through those two points.

Can an intercept be fraction or decimal?

Yes. Intercepts can be any real number. Here the x-intercept 7/3 is valid fraction (≈ 2.333...).

How can I check my x-intercept is correct?

Substitute (7/3, 0) into the equation: 6*(7/3) - 4*0 = 14, which gives 14 = 14, so it’s correct.

How to find intercepts if coefficients are messy?

Still set one variable to zero and solve for the other. Use fraction reduction and decimals if needed; convert to slope-intercept form if that makes solving easier.

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