A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable raised to the first power. When graphed, a linear equation always forms a straight line, like this example y equals 0.5x plus 1.
Linear equations can be written in several standard forms. The standard form is A x plus B y equals C, where A, B, and C are constants. The slope-intercept form is y equals m x plus b, where m is the slope and b is the y-intercept. The point-slope form is y minus y one equals m times x minus x one, useful when you know a point and the slope.
The slope measures how steep a line is, calculated as rise over run or the change in y divided by the change in x. A positive slope means the line goes up from left to right. The y-intercept is where the line crosses the y-axis, at the point zero comma b. In this example, the slope is 1.5 and the y-intercept is 2.
To solve a linear equation, we isolate the variable using inverse operations. For example, to solve 2x plus 3 equals 11, first subtract 3 from both sides to get 2x equals 8. Then divide both sides by 2 to get x equals 4. We can verify this graphically where the line y equals 2x plus 3 intersects the horizontal line y equals 11 at the point 4 comma 11.
Linear equations have many real-world applications including distance and speed problems, cost and pricing models, temperature conversions, and population growth. For example, the formula to convert Celsius to Fahrenheit is F equals nine-fifths C plus 32, which is a linear equation. At 0 degrees Celsius, water freezes at 32 degrees Fahrenheit, and at 100 degrees Celsius, water boils at 212 degrees Fahrenheit. Linear equations help us model and solve everyday problems efficiently.