A linear function is a mathematical relationship between two variables that creates a straight line when graphed. The general form is y equals m x plus b, where m is the slope representing the rate of change, and b is the y-intercept where the line crosses the y-axis.
To solve for y when x is given, follow these steps: Start with the linear equation, substitute the given x value, perform arithmetic operations, and calculate the result. For example, if y equals 2x plus 3 and x equals 5, substitute 5 for x to get y equals 2 times 5 plus 3, which equals 10 plus 3, giving us y equals 13.
To solve for x when y is given, follow these steps: Start with the linear equation, substitute the given y value, subtract b from both sides, divide both sides by m, and calculate the result. For example, if y equals 2x plus 3 and y equals 11, substitute 11 for y to get 11 equals 2x plus 3. Subtract 3 from both sides to get 8 equals 2x. Then divide both sides by 2 to get x equals 4.
Let's practice with two examples. Example 1: For y equals negative 3x plus 7, find y when x equals 2. Substituting, we get y equals negative 3 times 2 plus 7, which equals negative 6 plus 7, giving us y equals 1. Example 2: For y equals 0.5x minus 4, find x when y equals negative 1. Substituting, we get negative 1 equals 0.5x minus 4. Adding 4 to both sides gives us 3 equals 0.5x. Dividing by 0.5 gives us x equals 6.
To summarize what we have learned: Linear functions have the form y equals mx plus b. To find y when x is given, substitute the x value and calculate. To find x when y is given, substitute y, subtract b, and divide by m. Practice with different examples builds confidence, and these skills are essential for algebra and beyond.