How can I find the slope of a line from its equation?

Look for the pattern y = mx + b. The number multiplying x, m, is the slope. The constant b is the y-intercept: it tells you the y-value when x = 0, and the corresponding point is (0, b). [S1]

For example, in y = 3x − 7, the slope is 3 and the y-intercept is −7, or (0, −7). The minus sign belongs to the intercept here; it does not change the slope. In y = −(3/4)x − 4, the slope is −3/4, while the y-intercept is −4. [S1]

A quick check: point to the x-term and ask, “What number is multiplying x?” Include its sign and any fraction. That number is the slope.

Rearrange it so y is alone, then compare it with y = mx + b. For 2x + y = 5, subtract 2x from both sides:

2x + y = 5 y = −2x + 5

Now the coefficient of x is −2, so the slope is −2. The constant 5 is the y-intercept, not the slope. [S1]

Keep the equation balanced: whatever operation you use on one side, use on the other. As another example, 3x + 2y = 8 becomes 2y = −3x + 8 after subtracting 3x. Divide every term by 2: y = −(3/2)x + 4. The slope is −3/2, and the y-intercept is 4. This comparison works because the equation has been rewritten in slope-intercept form. [S1]

Write the rearranged equation one step at a time, and label the two parts only after y is isolated. In y = −(3/2)x + 4, the factor attached to x is the whole coefficient −3/2; the separate constant is 4. Don’t read the denominator or the constant as the slope by itself.

You can also check what the intercept means: substitute x = 0. In this example, y = −(3/2)(0) + 4 = 4, so the y-intercept is (0, 4). That check confirms the intercept, while the coefficient of x remains the slope. [S1]

If you want a second way to interpret the slope, it is the change in y divided by the change in x—often called rise over run. [S1] A negative slope means the coefficient is negative; you do not need to calculate two points just to read it from an equation already in slope-intercept form.

Use this short routine:

1. Copy the equation accurately, including signs and fractions.
2. If y is not alone, rearrange to isolate y.
3. Compare the result with y = mx + b.
4. Record m as the slope and b as the y-intercept.
5. If unsure about b, substitute x = 0 and check the resulting y-value. [S1]

Copyable working:

Equation: [your equation] Rearranged: y = [coefficient]x + [constant] Slope: [coefficient of x] Y-intercept: [constant], or (0, [constant])

For example, 4x + 2y = 10 gives 2y = −4x + 10, then y = −2x + 5. So the slope is −2 and the y-intercept is (0, 5). The method applies when you can rewrite the line as y = mx + b. [S1]