Q. what is the x intercept in y = mx + b?

Q: How do you find the x-intercept of the line y=mx+b?

Set y=0 and solve 0=mx+b. If m\neq 0, the x-intercept is x=-\frac{b}{m}. That's the point \left(-\frac{b}{m},0\right).

Q: What if m=0 (a horizontal line)?

If m=0 then y=b. If b=0 the line is the x-axis (every x is an intercept). If b\neq0 there is no x-intercept because y never equals 0.

Q: What if the line is vertical?

vertical line cannot be written as y=mx+b. It has equation x=c. The x-intercept is c if the line crosses the x-axis, i.e., the point (c,0).

Q: What if b=0?

If b=0 the equation is y=mx. The x-intercept is x=0, so the line passes through the origin (0,0).

Q: How do I get the x-intercept from two given points?

First compute the slope m=\frac{y_2-y_1}{x_2-x_1}. Then find b by substituting a point into y=mx+b. Finally compute x=-\frac{b}{m} (provided m\neq0).

Q: How do x- and y-intercepts help graph a line?

Plot the y-intercept (0,b) and the x-intercept \left(-\frac{b}{m},0\right) and draw the line through them. Two intercepts uniquely determine a straight line.

Answer

Set y=0: \(0 = mx + b\), so \(x = -\frac{b}{m}\quad (m \neq 0)\). If \(m = 0\) and \(b \neq 0\): no x-intercept. If \(m = 0\) and \(b = 0\): every x is an intercept.

Detailed Explanation

Find the x-intercept of the line y = mx + b — step-by-step

Recall the definition: the x-intercept is the x-coordinate of the point where the graph crosses the x-axis. On the x-axis the y-coordinate is 0. Therefore set y equal to 0 in the equation of the line.

\[0 = m x + b\]
Isolate the term containing x by moving b to the left side (subtract b from both sides). This yields:

\[-b = m x\]
Solve for x by dividing both sides by m. This step is valid only if m is not zero. The algebra gives:

\[x = -\frac{b}{m}\]
Write the x-intercept as an ordered pair (x, y). Since y = 0 at the intercept, the point is:

\[\left(-\frac{b}{m},\,0\right)\]
Special cases to note:
- If m is not zero, the x-intercept is exactly \( -\dfrac{b}{m} \) and the intercept point is \( \left(-\dfrac{b}{m},0\right) \).
- If m = 0, the line is horizontal: y = b. Then:
  - If b = 0 as well, the line is y = 0 (the x-axis), so every x is an x-intercept (infinitely many x-intercepts).
  - If b ≠ 0, the horizontal line y = b does not meet the x-axis, so there is no x-intercept.

Final answer: For m ≠ 0, the x-intercept is \( -\dfrac{b}{m} \); the intercept point is \( \left(-\dfrac{b}{m},0\right) \). For m = 0, handle as described in the special cases above.

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FAQs

How do you find the x-intercept of the line \(y=mx+b\)?

Set \(y=0\) and solve \(0=mx+b\). If \(m\neq 0\), the x-intercept is \(x=-\frac{b}{m}\). That's the point \(\left(-\frac{b}{m},0\right)\).

What if \(m=0\) (a horizontal line)?

If \(m=0\) then \(y=b\). If \(b=0\) the line is the x-axis (every x is an intercept). If \(b\neq0\) there is no x-intercept because \(y\) never equals 0.

What if the line is vertical?

vertical line cannot be written as \(y=mx+b\). It has equation \(x=c\). The x-intercept is \(c\) if the line crosses the x-axis, i.e., the point \((c,0)\).

What does the sign of \(-\frac{b}{m}\) tell me?

The sign indicates which side of the origin the intercept lies. If \(-\frac{b}{m}>0\) the x-intercept is positive (right side); if negative, it's left. Sign depends on the signs of \(m\) and \(b\).

What if \(b=0\)?

If \(b=0\) the equation is \(y=mx\). The x-intercept is \(x=0\), so the line passes through the origin \((0,0)\).

How do I get the x-intercept from two given points?

First compute the slope \(m=\frac{y_2-y_1}{x_2-x_1}\). Then find \(b\) by substituting a point into \(y=mx+b\). Finally compute \(x=-\frac{b}{m}\) (provided \(m\neq0\)).

Can the x-intercept be infinite or undefined?

If the line is vertical, the x-value is a constant (finite) and the y-intercept may be undefined. For horizontal lines with \(b\neq0\) there is no x-intercept; for \(b=0\) there are infinitely many x-intercepts.

How do x- and y-intercepts help graph a line?

Plot the y-intercept \((0,b)\) and the x-intercept \(\left(-\frac{b}{m},0\right)\) and draw the line through them. Two intercepts uniquely determine a straight line.

Find the x-intercept by setting y=0.
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