Q. Find the y-intercept of the line \( y = \frac{9}{20}x + \frac{8}{3} \).

Q: What is the y-intercept of y = \frac{9}{20}x + \frac{8}{3}?

The y-intercept is b = \frac{8}{3}, so the intercept point is (0,\frac{8}{3}) which equals 2.\overline{6}.

Q: How do you find the y-intercept from slope-intercept form?

Set x = 0; then y = b. For y = \frac{9}{20}x + \frac{8}{3}, y|_{x=0} = \frac{8}{3}.

Q: What is the x-intercept of the line?

Set y = 0 and solve: 0 = \frac{9}{20}x + \frac{8}{3} \Rightarrow x = -\frac{160}{27} \approx -5.9259.

Q: What are the coordinates of both intercepts?

y-intercept: (0,\frac{8}{3}). x-intercept: (-\frac{160}{27},0).

Q: How would you graph this line using slope and intercept?

Plot (0,\frac{8}{3}) then use rise/run = \frac{9}{20}: from that point go right 20 and up 9 for another point (or left 20 and down 9).

Q: How do you convert the equation to standard form Ax + By = C?

Multiply by 60 to clear denominators: 60y = 27x + 160, rearrange to 27x - 60y = -160.

Q: How can the y-intercept be written as a mixed number or decimal?

\frac{8}{3} = 2\frac{2}{3} = 2.6\overline{6}.

Answer

Set \( x = 0 \): \( y = \frac{9}{20}\cdot 0 + \frac{8}{3} = \frac{8}{3} \).

Y-intercept: \( \left(0, \frac{8}{3}\right) \).

Detailed Explanation

Recognize the form of the equation.

The equation is given as \( y = \tfrac{9}{20}x + \tfrac{8}{3} \). This is in slope–intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Identify the y-intercept directly from the form.

In the form \( y = mx + b \), the y-intercept is the constant term \( b \). Comparing with \( y = \tfrac{9}{20}x + \tfrac{8}{3} \), we see \( b = \tfrac{8}{3} \).
Verify by setting \( x = 0 \).

Another way is to evaluate the line at \( x = 0 \) because the y-intercept is the point where the graph crosses the y-axis (where \( x=0 \)). Compute:

\[ y = \tfrac{9}{20}\cdot 0 + \tfrac{8}{3} = 0 + \tfrac{8}{3} = \tfrac{8}{3}. \]
State the y-intercept as a point and a value.

The y-intercept value is \( \tfrac{8}{3} \). As a point on the coordinate plane, the y-intercept is \( \bigl(0,\tfrac{8}{3}\bigr) \). (Decimal form: \( \tfrac{8}{3} \approx 2.666\ldots \).)

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FAQs

What is the y-intercept of \(y = \frac{9}{20}x + \frac{8}{3}\)?

The y-intercept is \(b = \frac{8}{3}\), so the intercept point is \((0,\frac{8}{3})\) which equals \(2.\overline{6}\).

How do you find the y-intercept from slope-intercept form?

Set \(x = 0\); then \(y = b\). For \(y = \frac{9}{20}x + \frac{8}{3}\), \(y|_{x=0} = \frac{8}{3}\).

What is the slope of the line?

The slope is \(m = \frac{9}{20}\).

What is the x-intercept of the line?

Set \(y = 0\) and solve: \(0 = \frac{9}{20}x + \frac{8}{3} \Rightarrow x = -\frac{160}{27} \approx -5.9259\).

What are the coordinates of both intercepts?

y-intercept: \((0,\frac{8}{3})\). x-intercept: \((-\frac{160}{27},0)\).

How would you graph this line using slope and intercept?

Plot \((0,\frac{8}{3})\) then use rise/run = \(\frac{9}{20}\): from that point go right 20 and up 9 for another point (or left 20 and down 9).

How do you convert the equation to standard form Ax + By = C?

Multiply by 60 to clear denominators: \(60y = 27x + 160\), rearrange to \(27x - 60y = -160\).

What is the angle of inclination of the line?

\(\theta = \arctan\!\left(\frac{9}{20}\right) \approx 24.23^{\circ}\).

How can the y-intercept be written as a mixed number or decimal?

\(\frac{8}{3} = 2\frac{2}{3} = 2.6\overline{6}\).

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