Linear Equations

Understanding Linear Equations

A linear relationship changes by the same amount for each equal step in the input. This constant change is the key idea behind a straight line. For example, a taxi fare may begin with a fixed pickup charge, then increase by the same price for every mile.

The fixed charge is the starting value. The price per mile is the rate of change.

This pattern appears in phone plans, hourly pay, temperature conversions, savings accounts with regular deposits, and distance traveled at a constant speed. Real situations are not always perfectly linear, but a line is useful when the rate stays steady over the interval being studied.

To graph from slope-intercept form, plot the intercept first because it gives one exact point on the line. Then use the slope as a movement rule. A slope of three halves means move up three units and right two units.

A slope of negative three halves means move down three units and right two units. This movement is often called rise over run. You can reverse both directions and move down three, left two for a positive slope.

Both moves reach points on the same line. Plotting at least two points is enough to draw a line, but a third point is a useful check against small counting errors.

Equations are often given in forms that do not show the slope or starting value immediately. In that case, rearrange the equation until the output variable is alone. Use inverse operations carefully and apply each operation to every term on both sides.

For example, if two y minus six x equals ten, add six x to both sides, then divide every term by two. The result shows a slope of three and an intercept of five. A common mistake is dividing only one term instead of the entire side.

Another is losing a negative sign when moving terms. Writing each algebra step on its own line makes these errors easier to catch.

Tables provide another way to test whether a relationship is linear. Compare the changes in the output values when the input increases by equal amounts. If the output change is constant, the slope is constant.

If input steps are unequal, divide the change in output by the change in input before comparing rates. On a graph, pay attention to the scale on each axis. One square may represent one unit, five units, or much more.

A line can look steep or flat because of the chosen scale, even though its numerical slope has not changed. Check the units too.

A slope of sixty miles per hour means something different from a slope of sixty dollars per hour. Units tell what the rate actually describes.