Q. Find the slope of the line \(y = \frac{2}{3}x + \frac{3}{2}\).

Q: How do you identify the slope from slope–intercept form?

In y=mx+b, the coefficient m is the slope. For y=\frac{2}{3}x+\frac{3}{2}, m=\frac{2}{3}.

Q: What is the slope of a line perpendicular to this one?

The perpendicular slope is the negative reciprocal: -\frac{3}{2}.

Q: How do you write the equation given a point and this slope?

Use point–slope: y-y_1=\frac{2}{3}(x-x_1). Substitute your point (x_1,y_1) and simplify to desired form.

Q: How do you convert the slope \frac{2}{3} to a decimal or percent?

Decimal: \frac{2}{3}\approx0.6667. Percent: 0.6667\times100\%=66.67\%.

Answer

In slope-intercept form \(y = mx + b\) the slope is \(m\). Here \(y = \frac{2}{3}x + \frac{3}{2}\), so \(m = \frac{2}{3}\).

Slope = \(\frac{2}{3}\).

Detailed Explanation

Solution

Write the given equation in clear form: \(y = \frac{2}{3}x + \frac{3}{2}\).
Recall the slope-intercept form of a line: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Compare the given equation with \(y = mx + b\). The coefficient of \(x\) is the slope, so

Slope: \(m = \frac{2}{3}\).
Verification (optional): pick two points on the line. For \(x=0\), \(y=\frac{3}{2}\) so one point is \(\left(0,\frac{3}{2}\right)\). For \(x=3\),

\(y=\frac{2}{3}\cdot 3 + \frac{3}{2} = 2 + \frac{3}{2} = \frac{7}{2}\), so another point is \(\left(3,\frac{7}{2}\right)\).

Compute the slope between these points:

\(m = \dfrac{\frac{7}{2} – \frac{3}{2}}{3 – 0} = \dfrac{\frac{4}{2}}{3} = \dfrac{2}{3}\).

Final answer: the slope is \( \frac{2}{3} \).

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FAQs

What is the slope of the line \(y=\frac{2}{3}x+\frac{3}{2}\)?

The slope is \(m=\frac{2}{3}\).

How do you identify the slope from slope–intercept form?

In \(y=mx+b\), the coefficient \(m\) is the slope. For \(y=\frac{2}{3}x+\frac{3}{2}\), \(m=\frac{2}{3}\).

What does a slope of \(\frac{2}{3}\) mean graphically?

It means a rise of 2 units for every 3 units moved to the right: rise/run \(=\frac{2}{3}\).

What is the slope of a line parallel to this one?

Parallel lines share the same slope, so any parallel line has slope \(\frac{2}{3}\).

What is the slope of a line perpendicular to this one?

The perpendicular slope is the negative reciprocal: \(-\frac{3}{2}\).

How do you find the x-intercept of this line?

How do you write the equation given a point and this slope?

Use point–slope: \(y-y_1=\frac{2}{3}(x-x_1)\). Substitute your point \((x_1,y_1)\) and simplify to desired form.

How do you convert the slope \(\frac{2}{3}\) to a decimal or percent?

Decimal: \(\frac{2}{3}\approx0.6667\). Percent: \(0.6667\times100\%=66.67\%\).

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