Intercepts of y = mx + c

Statement

For a straight-line equation in the form \(y = mx + c\), the line crosses the axes at two important points called intercepts:

\[ \text{y-intercept: } (0,c), \quad \text{x-intercept: } \left(-\tfrac{c}{m}, 0\right) \quad (m \neq 0) \]

Why it’s true

• The y-intercept occurs when \(x=0\). Substituting into \(y=mx+c\) gives \(y=c\), so the point is \((0,c)\).
• The x-intercept occurs when \(y=0\). Solving \(0=mx+c\) gives \(x=-c/m\), so the point is \((-c/m,0)\).

Recipe (how to use it)

1. To find the y-intercept, set \(x=0\) in the equation and solve for \(y\).
2. To find the x-intercept, set \(y=0\) in the equation and solve for \(x\).
3. Plot these two points on the axes. They provide a quick way to sketch the line.

Spotting it

Whenever a line is written in the form \(y=mx+c\), the constant \(c\) gives the y-intercept immediately. The x-intercept requires solving a simple equation.

Common pairings

• Finding the equation of a line given intercepts.
• Checking parallel and perpendicular lines using gradient and intercept form.
• Intersection problems in simultaneous equations.

Mini examples

1. Equation: \(y=2x+3\). Intercepts: y-intercept \((0,3)\), x-intercept \((-3/2,0)\).
2. Equation: \(y=-x+4\). Intercepts: y-intercept \((0,4)\), x-intercept \((4,0)\).

Pitfalls

• Forgetting the negative: The x-intercept is \(-c/m\), not \(c/m\).
• Zero gradient case: If \(m=0\), the line is horizontal. It has no x-intercept unless \(c=0\).
• Dividing incorrectly: Always divide by \(m\), ensuring it’s non-zero.

Exam strategy

• Use intercepts to sketch a line quickly.
• Check intercepts to confirm solutions to simultaneous equations graphically.
• Remember: the intercepts are reliable anchor points for straight-line graphs.

Summary

The intercepts of \(y=mx+c\) give quick access to the points where the line meets the axes. The y-intercept is \((0,c)\), and the x-intercept is \((-c/m,0)\) provided \(m\neq0\). These intercepts are essential for sketching and solving problems involving straight lines.