A linear equation is a fundamental concept in algebra. It's called 'linear' because when we graph it on a coordinate plane, it forms a straight line. The key characteristic is that all variables have an exponent of exactly one. Here we see the equation y equals 2x plus 1, which creates this straight line passing through points like (0,1) and (1,3).
Linear equations have standard forms depending on the number of variables. For one variable, the general form is ax plus b equals zero, where a cannot be zero. For two variables, it's ax plus by equals c, where a and b cannot both be zero. These forms help us identify and work with linear relationships systematically.
The slope-intercept form y equals mx plus b is the most common way to write linear equations. Here, m represents the slope, which measures how steep the line is. The slope is calculated as rise over run. The b represents the y-intercept, where the line crosses the y-axis. In our example, the slope is 1.5 and the y-intercept is 1.
To solve a linear equation like 2x plus 3 equals 7, we use inverse operations to isolate the variable. First, we subtract 3 from both sides to get 2x equals 4. Then we divide both sides by 2 to get x equals 2. We can verify this graphically by finding where the line y equals 2x plus 3 intersects with y equals 7, which occurs at the point (2, 7).
Linear equations have countless real-world applications across many fields. In economics, they model cost and revenue relationships. In physics, they describe motion and forces. In business, they help with pricing and budgeting decisions. For example, a phone plan might cost twenty dollars per month plus ten cents per minute, giving us the linear equation C equals 20 plus 0.1m, where C is the total cost and m is the number of minutes used.