A linear equation is a mathematical equation where variables are raised to the first power only. The general form is ax plus b equals c. For example, 2x plus 3 equals 7. Key characteristics include variables to the first power only, no multiplication between variables, and their graphs form straight lines. When we graph the equation 2x plus 3 equals 7, we get a straight line, and the solution x equals 2 corresponds to the point where the line intersects.
Linear equations can be written in three main standard forms. The standard form is Ax plus By equals C, where A, B, and C are coefficients. The slope-intercept form is y equals mx plus b, where m is the slope and b is the y-intercept. The point-slope form is y minus y1 equals m times x minus x1, using a known point and slope. Each form highlights different components of the line, and the same line can be expressed in all three forms.
To solve one-variable linear equations, we use algebraic operations to isolate the variable. Let's solve 3x plus 5 equals 14. First, subtract 5 from both sides to get 3x equals 9. Then divide both sides by 3 to get x equals 3. The balance scale visualization shows that whatever operation we perform on one side of the equation, we must perform on the other side to maintain equality.
There are multiple methods to graph linear equations. First, we can plot points from a table of values. Second, we can use the slope and y-intercept method. Third, we can find the x and y intercepts. For the equation y equals 2x minus 1, the slope is 2 and y-intercept is negative 1. We can plot points like zero comma negative one, one comma one, and two comma three, then connect them to form a straight line.
Systems of linear equations involve two or more equations working together. For example, x plus y equals 3 and 2x minus y equals 0. We can solve using three methods: graphing to find intersection points, substitution, or elimination. The graphical method shows two lines intersecting at point one comma two, which is our unique solution. Systems can have one solution, no solution if lines are parallel, or infinite solutions if lines are identical.