Linear equations describe constant rates of change, so their graphs are straight lines. They are used to model motion, cost, temperature change, income, and many other relationships where one quantity changes steadily with another. Different forms of a linear equation show different information quickly.
Learning to move between forms helps you graph lines, compare relationships, and solve problems efficiently.
The three most common forms are slope-intercept form, point-slope form, and standard form. Slope-intercept form makes the slope and y-intercept easy to see, point-slope form is useful when you know one point and the slope, and standard form is helpful for intercepts and systems of equations. All three can represent the same line, so converting between them is mainly algebraic rearrangement.
A good strategy is to identify what information is given, choose the form that uses it most directly, and then convert only if needed.
Key Facts
- Slope-intercept form: y = mx + b, where m is slope and b is the y-intercept.
- Point-slope form: y - y1 = m(x - x1), where m is slope and (x1, y1) is a point on the line.
- Standard form: Ax + By = C, where A, B, and C are usually integers and A is often written as nonnegative.
- Slope formula: m = (y2 - y1)/(x2 - x1).
- To graph y = mx + b, plot (0, b), then use rise/run from the slope m.
- To find intercepts from Ax + By = C, set y = 0 for the x-intercept and set x = 0 for the y-intercept.
Vocabulary
- Slope
- Slope is the constant rate of change of a line, found by dividing vertical change by horizontal change.
- Y-intercept
- The y-intercept is the point where a graph crosses the y-axis, usually written as (0, b).
- X-intercept
- The x-intercept is the point where a graph crosses the x-axis, where y equals 0.
- Linear equation
- A linear equation is an equation whose graph is a straight line and whose variables have no exponents other than 1.
- Equivalent equations
- Equivalent equations are equations that have the same solution set or represent the same graph.
Common Mistakes to Avoid
- Mixing up slope and y-intercept in y = mx + b. The coefficient of x is the slope, while the constant term is the y-value where the line crosses the y-axis.
- Using the slope formula in the wrong order. If you start with y2 - y1 in the numerator, you must use x2 - x1 in the denominator in the same point order.
- Forgetting to distribute the slope in point-slope form. In y - y1 = m(x - x1), the m multiplies every term inside the parentheses.
- Changing the graph when converting forms. Legal algebra operations keep the same line, but arithmetic errors or sign mistakes can create a different equation.
Practice Questions
- 1 Write the equation of the line with slope 3 and y-intercept -4 in slope-intercept form, then convert it to standard form.
- 2 A line passes through (2, 5) and (6, 13). Find its slope, write the equation in point-slope form, and then write it in slope-intercept form.
- 3 A student knows the x-intercept and y-intercept of a line but not its slope. Which form of a linear equation is most useful to start with, and how could the student graph the line from that information?