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

Answer

Set y to 0.
\[ 3x + 6(0) = 21 \]
Solve for x.
\[ 3x = 21 \]

\[ x = 7 \]
State the x-intercept.
\[ (7, 0) \]

Detailed Explanation

Understand what the x-intercept is:
The x-intercept is the point where the graph crosses the x-axis, which means the y-coordinate is 0. We start with the given line:

\(3x + 6y = 21\)
Set y to 0 and substitute:
Replace y with 0 because at the x-intercept y = 0:

\(3x + 6(0) = 21\)
Simplify the equation:
Multiply out 6(0) = 0 and simplify the left side:

\(3x + 0 = 21\) which simplifies to \(3x = 21\)
Solve for x:
Divide both sides of \(3x = 21\) by 3 to isolate x:

\(x = \dfrac{21}{3} = 7\)
State the x-intercept as an ordered pair:
The x-intercept is \((7, 0)\).

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Frequently Asked Questions

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

Set y = 0 and solve: 3x = 21 so x = 7. The x-intercept is the point (7, 0).

How do you find an x-intercept in general?

For any line, set y = 0 and solve the resulting equation for x. The solution gives the x-coordinate of the intercept; the point is (x, 0).

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

Set x = 0: 6y = 21, so y = 21/6 = 7/2. The y-intercept is (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-intercept form is y = mx + b with m = -1/2 and b = 7/2.

What is the slope of this line?

From y = (-1/2)x + 7/2, the slope m = -1/2.

How do I graph the line quickly using intercepts?

Plot the x-intercept (7, 0) and y-intercept (0, 7/2). Draw straight line through these two points.

Could this line have no x-intercept?

Only vertical lines (x = constant) lack y-intercept; only horizontal lines (y = constant) lack unique x-intercept? More correctly: nonvertical line always has one x-intercept; vertical lines have either no x-intercept (if x ≠ 0) or infinitely many if x = 0 coincides with origin for y=0.

How can I check my x-intercept is correct?

Substitute (7, 0) into the original equation: 3(7) + 6(0) = 21, which is true, so (7, 0) is correct.

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