Linear equations describe relationships that change at a constant rate. This cheat sheet helps students connect tables, graphs, equations, and real-world situations. It is useful for quickly identifying slope, intercepts, and the correct equation form.
Students in grades 7-9 use these skills often in algebra, coordinate graphing, and word problems.
The most important idea is that slope measures the rate of change between two points. The slope formula is , and slope-intercept form is . Point-slope form, , is useful when a point and slope are known.
Standard form, , helps with intercepts and comparing linear equations.
Key Facts
- The slope between two points is , where the numerator is the vertical change and the denominator is the horizontal change.
- Slope-intercept form is , where is the slope and is the -intercept.
- Point-slope form is , which uses a known point and slope .
- Standard form is , where , , and are constants and is usually nonnegative.
- A horizontal line has slope and an equation like .
- A vertical line has undefined slope and an equation like .
- Parallel lines have equal slopes, so if two nonvertical lines are parallel, then .
- Perpendicular lines have slopes whose product is , so when neither line is vertical or horizontal.
Vocabulary
- Slope
- Slope is the rate of change of a line, calculated by .
- Y-intercept
- The -intercept is the point where a graph crosses the -axis, usually written as .
- X-intercept
- The -intercept is the point where a graph crosses the -axis, found by setting .
- Linear equation
- A linear equation is an equation whose graph is a straight line and has a constant rate of change.
- Slope-intercept form
- Slope-intercept form is , where the slope and -intercept can be read directly.
- Point-slope form
- Point-slope form is and is used when one point and the slope are known.
Common Mistakes to Avoid
- Using for slope: This reverses rise and run, so the slope becomes the reciprocal of the correct value.
- Mixing the order of points in the slope formula: If you start with in the numerator, you must use in the denominator.
- Confusing the -intercept with the slope in : The coefficient is the slope, while is the value of when .
- Treating vertical lines as having slope : A vertical line has undefined slope because its run is , which would require division by .
- Forgetting to distribute in point-slope form: In , the slope must multiply both terms inside the parentheses.
Practice Questions
- 1 Find the slope of the line through and .
- 2 Write the equation of the line with slope and -intercept in slope-intercept form.
- 3 A line passes through with slope . Write its equation in point-slope form and then slope-intercept form.
- 4 Explain how you can tell from a graph whether a line has positive slope, negative slope, zero slope, or undefined slope.
Understanding Linear Equations & Slope
A line on a coordinate grid represents every pair of values that follows one rule. To calculate its steepness reliably, keep the subtraction order consistent. If you subtract the first horizontal value from the second horizontal value, do the same with the vertical values.
Reversing both orders gives the same result because both changes reverse signs. Reversing only one order creates a wrong sign. Reduce the resulting fraction when possible.
A slope of three halves means the line goes up three units for every two units moved right. A negative slope means that higher horizontal values match lower vertical values.
When you need an equation from a slope and one point, use the known point to find the starting value on the vertical axis. Put that point into the slope-intercept equation, then solve for the intercept. This process is more dependable than trying to estimate the intercept from a rough graph.
Check the finished equation by using a second known point. The point-slope form is especially useful before the equation has been simplified. Be careful when multiplying through parentheses, particularly with negative slopes.
Standard form can be created by moving variable terms to one side and the constant term to the other. Different-looking equations may describe exactly the same line.
Linear equations are common in situations with a fixed starting amount and a steady change per unit. A taxi fare can include an initial charge plus a charge for each mile. A savings account can begin with some money, then grow by the same weekly deposit.
In these cases, the intercept has a real meaning as the amount at zero miles or zero weeks. The slope carries units. If distance is measured in miles and time in hours, the slope is miles per hour.
Units help reveal mistakes. A slope of five dollars per hour cannot sensibly be used as five hours per dollar.
Not every pattern is linear. Changing speeds, sale discounts, and population growth often fail to stay at one constant rate.
Line relationships provide useful shortcuts on graphs. Parallel lines keep the same tilt, so they never meet unless they are actually the same line. Perpendicular lines meet at a right angle.
For slanted lines, their slopes have opposite signs and reciprocal sizes. Vertical and horizontal lines need special care because the usual reciprocal rule does not apply in the same way. Graph scale matters too.
A line can look steep or flat when the horizontal and vertical axes use different unit sizes. Always read the numbered scale before judging a graph. Use intercepts, two points, and the direction of the line as separate checks that your equation matches the graph.