A linear equation is a fundamental concept in algebra. It's called 'linear' because when we graph it, we get a straight line. The key characteristic is that all variables have an exponent of exactly one. For example, y equals 2x plus 1 is a linear equation. Notice how the points lie perfectly on a straight line.
Linear equations can be written in several standard forms. For one variable, we have ax plus b equals zero. For two variables, the general form is ax plus by plus c equals zero. The most common form is slope-intercept form: y equals mx plus b, where m is the slope and b is the y-intercept. Here we see three different linear equations, each forming a straight line with different slopes.
In the slope-intercept form y equals mx plus b, m represents the slope and b represents the y-intercept. The slope tells us how steep the line is - it's calculated as rise over run. The y-intercept is where the line crosses the y-axis. In this example, the slope is 1.5, meaning for every 2 units we move right, we go up 3 units. The y-intercept is 1, so the line crosses the y-axis at the point zero, one.
To solve a linear equation, we use inverse operations to isolate the variable. Let's solve 3x plus 5 equals 14. First, we subtract 5 from both sides to get 3x equals 9. Then we divide both sides by 3 to get x equals 3. We can verify this by substituting back: 3 times 3 plus 5 equals 14, which is correct. Graphically, this represents finding where the line y equals 3x plus 5 intersects with the horizontal line y equals 14.
Linear equations have countless real-world applications. They model relationships like distance equals speed times time, cost equals rate times quantity, and temperature conversions. In business, they help calculate profit models and pricing strategies. This example shows a service cost model where the total cost equals 10 dollars per hour plus a 20 dollar fixed fee. Linear equations are fundamental tools for understanding and predicting relationships in science, economics, and everyday situations.