LEMMA

Lines & systems

Read linear graphs

ALEKS placement

Read the intercept where the line crosses the y-axis, then count rise over run.

What this covers

  • Find the y-intercept: the y-value where x = 0.
  • Pick a second point the line passes through exactly.
  • Count vertical change over horizontal change between them.
  • Assemble y = mx + b.

Worked example

Worked example

A line crosses the y-axis at 4 and passes through (3, -2). Write its equation.

  1. m=3−0−2−4​Use the intercept (0, 4) as the first point and (3, -2) as the second. Subtract in that order on top and bottom.
  2. m=−2-6 over 3 is -2, so the line falls.
  3. y=−2x+4Substitute m and b into slope-intercept form.
Another worked example

Graph 2x + 3y = 6 using intercepts.

  1. x=3For the x-intercept, set y to zero. Then two x equals six, so the graph meets the x-axis at the point three, zero.
  2. y=2For the y-intercept, set x to zero. Then three y equals six, so the graph meets the y-axis at the point zero, two.
  3. y=−32​x+2From the x-intercept to the y-intercept the run is negative three and the rise is two, so the slope is negative two thirds and the y-intercept is two.

A common mistake

A mistake Lemma catches

From

m=3−04−(−2)​m=3−0−2−4​

“The line runs from (0, 4) down to (3, −2), so it falls and its slope is negative. The top used 4 − (−2), first point minus second, while the bottom used 3 − 0, second minus first; mismatched orders give 6/3 = 2. Subtract in one order: (−2 − 4)/(3 − 0) = −6/3 = −2.”

Try one

Sample problem

At what y-value does the plotted line cross the y-axis?

Coordinate graph of a straight line through (-2, -5) and (2, 3).

Practice linear graphs free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

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