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Learn to extract k from proportional lines by computing y/x at marked points, writing y = kx, and checking with slope triangles.
A proportional relationship graphs as a straight line through the origin. The constant of proportionality k is the ratio y/x for any point on the line (except the origin). This lesson shows how to read k from plotted lines, write the equation y = kx, and verify your value using a slope triangle. You will practice with marked points and simple arithmetic, using consistent units and clearly labeled axes.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
A proportional relationship has a graph that is a straight line passing through the origin (0,0). If the line does not pass through the origin, it is not proportional, and k is not defined in this lesson.
Each graph plate shows one line with at least one marked point (x, y). The coordinates are read from the axes, which have consistent units. Choose a marked point other than the origin to compute k.
For a proportional line, k equals y divided by x for any point on the line except the origin. Use the marked point to compute k. Keep units consistent; if x is in hours and y is in dollars, k is dollars per hour.
After computing k, write the equation in the form y = kx. Substitute the marked point into the equation to check that both sides match.
A slope triangle uses two points on the line to show rise over run. For a proportional line, rise/run equals k. Draw a right triangle with horizontal leg run = x2 − x1 and vertical leg rise = y2 − y1. The ratio rise/run should match y/x from any point.
Use the origin and the marked point for a simple triangle: run = x, rise = y, so rise/run = y/x = k. This confirms your value without relying on a single calculation.
k may be a fraction or decimal. Keep the exact fraction if possible, then convert to a decimal only if needed. Label k with units derived from the axes, such as meters per second or cost per item.
When axes use different scales, read coordinates carefully. The value of k does not depend on the scale as long as you read both coordinates correctly. Always state units in your final answer.
A graph shows a straight line through the origin with a marked point at (4, 10). The x-axis is in hours and the y-axis is in dollars. Find k, write the equation, and verify with a slope triangle.
k = 10/4 = 2.5 dollars per hour. Equation: y = 2.5x. Slope triangle from (0,0) to (4,10): run = 4, rise = 10, rise/run = 10/4 = 2.5, which matches k.
A proportional line passes through the origin and has a marked point at (3, 12). The x-axis is in seconds and the y-axis is in meters. Find k and write the equation.
k = 12/3 = 4 meters per second. Equation: y = 4x.
Why: For a proportional line, k = y/x. Using (3,12), k = 12/3 = 4. The equation is y = 4x with units meters per second.
Check: Substitute (3,12) into y = 4x: 12 = 4(3) = 12. Also draw a slope triangle from (0,0) to (3,12): rise 12, run 3, ratio 4.
A graph shows a line through the origin with a marked point at (5, 7). The axes have no units. Compute k as a fraction and as a decimal rounded to two places. Write the equation.
k = 7/5 = 1.4. Equation: y = 1.4x (or y = (7/5)x).
Why: k = y/x = 7/5. The fraction simplifies to 7/5, which equals 1.4. The equation uses this constant.
Check: Substitute (5,7): 7 = 1.4(5) = 7. Slope triangle from origin: rise 7, run 5, ratio 7/5 = 1.4.
A proportional line through the origin has a marked point at (2, 9). The x-axis is in items and the y-axis is in dollars. Find k, write the equation, and state the unit of k.
k = 9/2 = 4.5 dollars per item. Equation: y = 4.5x.
Why: k = y/x = 9/2 = 4.5. Units are dollars divided by items, so dollars per item. The equation is y = 4.5x.
Check: Substitute (2,9): 9 = 4.5(2) = 9. Slope triangle: rise 9, run 2, ratio 4.5.
A line through the origin has a marked point at (8, 6). Another point on the line is (4, 3). Compute k using both points and confirm they agree. Write the equation.
Using (8,6): k = 6/8 = 3/4 = 0.75. Using (4,3): k = 3/4 = 0.75. Equation: y = 0.75x (or y = (3/4)x).
Why: Both points give the same ratio y/x = 3/4. Since the line passes through the origin, k is constant. The equation uses k = 3/4.
Check: Substitute each point into y = 0.75x: 6 = 0.75(8) = 6 and 3 = 0.75(4) = 3. Slope triangle between points: run = 8−4 = 4, rise = 6−3 = 3, ratio 3/4 = 0.75.
Yes, if the line passes through the origin and has a negative slope, k is negative. Compute y/x using a marked point with a negative coordinate.
The origin gives 0/0, which is undefined. Choose a different marked point with x not equal to zero.
Substitute the marked point into y = kx. If both sides are equal, your k is consistent with that point. You can also use a slope triangle from the origin to the point.
Simplifying helps compare values and write cleaner equations. Keep the exact value; decimals are acceptable if stated clearly.