Carcass wrote:
In the xy-plane, find the following.
(a) Slope and y-intercept of the line with equation 2y+x=6
(b) Equation of the line passing through the point (3,2) with y-intercept 1
(c) The y-intercept of a line with slope 3 that passes through the point (−2,1)
(d) The x-intercepts of the graphs in (a), (b), and (c)
(a) Slope: \(- \frac{1}{2}\); y-intercept:; 3 (b) \(y=\frac{x}{3}+1\) (c) 7 (d) \(6,-3, - \frac{7}{3}\)
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 7By 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 6b) 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 -3c) 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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Brent Hanneson – Creator of greenlighttestprep.com
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