It's often useful to take the equation of a line and rewrite it slope y-intercept form y = mx + b, where m is the line's slope, and b is the line's y-intercept.

(a) Slope and y-intercept of the line with equation \(2y + x = 6\)

Take: \(2y + x = 6\)

Subtract x from both sides: \(2y = -x + 6\)

Divide both sides by 2 to get: \(y = -\frac{1}{2}x + 3\)

We can see that the slope is \(-\frac{1}{2}\), and the y-intercept is 3
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(b) Equation of the line passing through the point (3,2) with y-intercept 1

If the y-intercept is 1, then \(b = 1\)

So, far we have: \(y = mx + 1\)

Since the point (3,2) lies ON the line, it's coordinates (x = 3 and y = 2) must SATISFY the equation of the line.

Replace values to get: \(2 = m(3) + 1\)

Subtract 1 from both sides to get: \(1 = 3m\)

Solve: \(m = \frac{1}{3}\)

The equation of the line is \(y = \frac{1}{3}x + 1\)
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(c) The y-intercept of a line with slope 3 that passes through the point (−2,1)

If the slope is 3, then \(m = 3\)

So, far we have: \(y = 3x + b\)

Since the point (−2,1) lies ON the line, it's coordinates (x = -2 and y = 1) must SATISFY the equation of the line.

Replace values to get: \(1 = 3(-2) + b\)

Simplify: \(1 = -6 + b\)

Solve: \(b = 7\)

So, the y-intercept is 7

By the way, the equation of the line is \(y = 3x + 7\)
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(d) The x-intercepts of the graphs in (a), (b), and (c)
Key concept: the x-intercept is the x-value when y = 0

So, for each equation, replace y with 0 and solve for x.

a) Replace y with 0 to get: \(0 = -\frac{1}{2}x + 3\)

Subtract 3 from both sides: \(-3 = -\frac{1}{2}x\)

Solve: \(x = 6\)

The x-intercept is 6


b) Replace y with 0 to get: \(0 = \frac{1}{3}x + 1\)

Subtract 1 from both sides: \(-1 = \frac{1}{3}x\)

Solve: \(x = -3\)

The x-intercept is -3



c) Replace y with 0 to get: \(0 = 3x + 7\)

Subtract 7 from both sides: \(-7 = 3x\)

Solve: \(x = -\frac{7}{3}\)

The x-intercept is \(-\frac{7}{3}\)

Cheers,
Brent
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