Find equation given slope and y-intercept with examples.

Equation of Line in Slope-Intercept Form y = mx + c

Question

A line has slope $m = 3$ and y-intercept $c = -5$ . Write the equation of the line in slope-intercept form. Also find the x-intercept.

Solution — Step by Step

The slope-intercept form is $y = mx + c$ , where $m$ is the slope and $c$ is the y-intercept. We have $m = 3$ and $c = -5$ , so:

y = 3x - 5

That’s our equation. No rearranging needed — this form is ready to use.

At the y-intercept, $x = 0$ . Substituting: $y = 3(0) - 5 = -5$ .

The point $(0, -5)$ lies on the line — matches our given y-intercept, so we’re correct.

At the x-intercept, $y = 0$ . Set $y = 0$ and solve for $x$ :

0 = 3x - 5 \implies 3x = 5 \implies x = \frac{5}{3}

The x-intercept is $\left(\dfrac{5}{3},\ 0\right)$ .

Equation of line: $y = 3x - 5$
x-intercept: $\left(\dfrac{5}{3},\ 0\right)$
y-intercept: $(0, -5)$ (given)

Why This Works

The slope-intercept form $y = mx + c$ is designed so you can read off two pieces of information at a glance. The coefficient of $x$ is always the slope, and the constant term is always the y-intercept. No algebra required — it’s built into the structure of the form.

The slope $m = 3$ means for every 1 unit we move right along the x-axis, the line rises 3 units. A positive slope means the line goes up left to right. Since $c = -5$ is negative, the line cuts the y-axis below the origin.

This is the most tested form in Class 11 board exams and JEE Main coordinate geometry. CBSE 2024 Board Exam directly asked students to write the equation given $m$ and $c$ — exactly this pattern. Knowing this form cold saves time in both MCQs and long-answer questions.

Alternative Method — Using Two Points

If you’re given $m$ and $c$ , you can also use the two-point form as a check.

We already know two points: $(0, -5)$ and $\left(\dfrac{5}{3}, 0\right)$ .

The two-point form is:

\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}

Substituting $(x_1, y_1) = (0, -5)$ and $(x_2, y_2) = \left(\dfrac{5}{3}, 0\right)$ :

\frac{y - (-5)}{x - 0} = \frac{0 - (-5)}{\frac{5}{3} - 0} = \frac{5}{\frac{5}{3}} = 3

y + 5 = 3x \implies y = 3x - 5

Same answer. This cross-check is useful in board exams when you want to confirm your equation before writing the final answer.

Common Mistake

Students often confuse the sign of $c$ . If the equation is $y = 3x - 5$ , the y-intercept is $-5$ , NOT $+5$ . The formula is $y = mx + c$ , so $c$ is what’s added. Here $c = -5$ because we’re adding $(-5)$ . Writing the intercept as $5$ when the line crosses at $(0, -5)$ is a direct mark loss in boards.

For finding x-intercept quickly: set $y = 0$ in the equation and solve. For y-intercept: set $x = 0$ . This works for any line equation, not just slope-intercept form — keep this as a reflex, not a formula to remember.

Question

Solution — Step by Step

Why This Works

Alternative Method — Using Two Points

Common Mistake

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