Q. How to find the y-intercept with slope and a point.

Q: How do I find the y-intercept when you know the slope and one point on the line?

Use point-slope then solve for b. From y - y_1 = m(x - x_1) rearrange to y = mx + b. So b = y_1 - m x_1.

Q: What if the given point is already on the y-axis?

If the point is (0,y_0) then the y-intercept is y_0. No calculation needed: b = y_0.

Q: How do I write the full equation of the line from slope and a point?

After finding b = y_1 - m x_1, plug into y = mx + b. The line is y = m x + (y_1 - m x_1).

Q: What if the slope is zero?

If m=0 the line is horizontal and y=b. Use the point (x_1,y_1) to get b=y_1.

Q: Can I find the y-intercept from two points?

Yes. First compute slope m = (y_2 - y_1)/(x_2 - x_1). Then use b = y_1 - m x_1 (or y_2 - m x_2).

Q: How do I find the x-intercept after getting the y-intercept?

Set y=0 in y = m x + b and solve: x = -b/m (if m \neq 0). If m=0, there is no x-intercept unless b=0.

Q: How do I find the y-intercept from standard form Ax + By = C?

Solve for y: y = -(A/B)x + C/B. The y-intercept is b = C/B provided B \neq 0.

Answer

Start from the slope–intercept form: \(y = mx + b\).
Given slope \(m\) and point \((x_1,y_1)\), substitute: \(y_1 = m x_1 + b\), so \(b = y_1 – m x_1\).

Example: \(m = 2\), point \((3,5)\) \(\rightarrow\) \(b = 5 – 2\cdot 3 = -1\), equation \(y = 2x – 1\).

Detailed Explanation

How to find the y-intercept when you are given the slope and one point

Write the equation of the line in slope-intercept form.

The slope-intercept form of a line is

\[y = m x + b\]
where \(m\) is the slope and \(b\) is the y-intercept (the value of \(y\) when \(x = 0\)).
Substitute the given point and slope into the equation.

If the given point is \((x_1, y_1)\) and the slope is \(m\), replace \(x\) by \(x_1\) and \(y\) by \(y_1\) in the slope-intercept equation. This gives

\[y_1 = m x_1 + b.\]
Solve that equation for \(b\) (the y-intercept).

Isolate \(b\) by subtracting \(m x_1\) from both sides. The algebraic steps are:

\[y_1 = m x_1 + b\]

Subtract \(m x_1\) from both sides to obtain

\[y_1 – m x_1 = b.\]

Therefore the y-intercept is

\[b = y_1 – m x_1.\]
Optionally, write the final equation of the line.

Once you compute \(b\), substitute it back into the slope-intercept form to get

\[y = m x + b,\]
where \(b\) is \(y_1 – m x_1\).
Special note about vertical lines.

If the slope is undefined (a vertical line), the slope-intercept form does not apply and the line has an equation of the form \(x = a\). A vertical line generally does not have a y-intercept unless \(a = 0\).

Worked example (to illustrate the steps)

Given slope \(m = 2\) and point \((3,5)\):

Substitute into \(b = y_1 – m x_1\):
\[b = 5 – 2 \cdot 3\]
\[b = 5 – 6\]
\[b = -1\]
The y-intercept is \(-1\), and the line is \[y = 2x – 1.\]

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FAQs

How do I find the y-intercept when you know the slope and one point on the line?

Use point-slope then solve for b. From \(y - y_1 = m(x - x_1)\) rearrange to \(y = mx + b\). So \(b = y_1 - m x_1\).

What if the given point is already on the y-axis?

If the point is \((0,y_0)\) then the y-intercept is \(y_0\). No calculation needed: \(b = y_0\).

How do I write the full equation of the line from slope and a point?

After finding \(b = y_1 - m x_1\), plug into \(y = mx + b\). The line is \(y = m x + (y_1 - m x_1)\).

What if the slope is zero?

If \(m=0\) the line is horizontal and \(y=b\). Use the point \((x_1,y_1)\) to get \(b=y_1\).

Can I find the y-intercept from two points?

Yes. First compute slope \(m = (y_2 - y_1)/(x_2 - x_1)\). Then use \(b = y_1 - m x_1\) (or \(y_2 - m x_2\)).

What happens with a vertical line?

How do I find the x-intercept after getting the y-intercept?

Set \(y=0\) in \(y = m x + b\) and solve: \(x = -b/m\) (if \(m \neq 0\)). If \(m=0\), there is no x-intercept unless \(b=0\).

How do I find the y-intercept from standard form \(Ax + By = C\)?

Solve for \(y\): \(y = -(A/B)x + C/B\). The y-intercept is \(b = C/B\) provided \(B \neq 0\).

Use the slope and a point to get b.
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