Q. Find the x-intercept of the line (\(9x – 3y = 24\)).
Answer
Set \(y = 0\):
\[ 9x – 3(0) = 24 \]
\[ 9x = 24 \]
\[ x = \frac{24}{9} = \frac{8}{3} \]
Final result: \(\boxed{\left(\frac{8}{3}, 0\right)}\)
Detailed Explanation
Problem: Find the x-intercept of the line \(9x-3y=24\).
Step 1 — Use the definition of an x-intercept
The x-intercept is the point where the line crosses the x-axis, so \(y=0\).
Step 2 — Substitute \(y=0\) into the equation
\[ 9x-3(0)=24 \]
Step 3 — Simplify
\[ 9x=24 \]
Step 4 — Solve for \(x\)
\[ x=\frac{24}{9} \]
Step 5 — Reduce the fraction
Both 24 and 9 are divisible by 3:
\[ 24\div 3=8, \qquad 9\div 3=3 \]
so
\[ x=\frac{8}{3} \]
Answer
The x-intercept is \(x=\tfrac{8}{3}\), which as a point is \(\bigl(\tfrac{8}{3}, \, 0\bigr)\).
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FAQs
How do I find the x-intercept of the line (9x-3y=24)?
Set (y=0) and solve: (9x-3(0)=24), so (9x=24) and (x=frac{24}{9}=frac{8}{3}). The x-intercept is (left(frac{8}{3},0right)).
How do I find the y-intercept?
Set (x=0): (9(0)-3y=24) so (-3y=24) and (y=-8). The y-intercept is ((0,-8)).
What is the slope of the line?
Rewrite to slope-intercept form: (9x-3y=24), (3y=9x-24), (y=3x-8). The slope is (3).
How can I simplify the equation first?
Divide every term by (3): (3x-y=8). This simpler form makes intercepts or slope easier to find.
How can I check the x-intercept is correct?
Substitute (x=frac{8}{3}) and (y=0): (9left(frac{8}{3}right)-3(0)=24), which gives (24=24), so the point is correct.
How to write the line in intercept form?
Intercept form is ( frac{x}{a}+frac{y}{b}=1). For this line (a=frac{8}{3}) and (b=-8), so ( frac{x}{8/3}+frac{y}{-8}=1).
How do I graph the line using intercepts?
Plot the intercepts (left(frac{8}{3},0right)) and ((0,-8)), then draw the straight line through them. Two points determine the line.
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