Welcome to learning about two-variable equations. A two-variable equation is an equation that contains two unknown variables, usually represented by x and y. The general form is A x plus B y equals C, where A, B, and C are constants. For example, two x plus three y equals twelve is a two-variable equation. When we graph this equation, we get a straight line, and every point on this line represents a solution to the equation.
Now let's learn how to find solutions to two-variable equations. A solution is a pair of values, x and y, that makes the equation true when substituted. The method is simple: choose a value for one variable, then solve for the other. Let's use the example x plus y equals five. If x equals zero, then y equals five, giving us the solution zero comma five. If x equals two, then y equals three, giving us two comma three. If x equals five, then y equals zero, giving us five comma zero. We can organize these solutions in a table showing the x values, y values, and the coordinate pairs.
Now let's learn about graphing two-variable equations. The graph of a linear two-variable equation is always a straight line on the coordinate plane. Every point on this line represents a solution to the equation. Let's graph the equation y equals two x minus one. We can plot several points by choosing x values and calculating the corresponding y values. When x is zero, y is negative one. When x is one, y is one. When x is two, y is three. When x is three, y is five. We plot these points and connect them with a straight line to complete our graph.
An important form for two-variable equations is the slope-intercept form: y equals m x plus b. In this form, m represents the slope, which tells us how steep the line is, and b represents the y-intercept, which is where the line crosses the y-axis. Let's look at the example y equals three x plus two. Here, the slope is three, meaning the line rises three units for every one unit to the right. The y-intercept is two, so the line crosses the y-axis at the point zero comma two. This form makes it easy to quickly graph a line by starting at the y-intercept and using the slope to find other points.
To summarize what we've learned about two-variable equations: They contain two unknown variables, usually x and y. Solutions are coordinate pairs that make the equation true when substituted. The graphs of linear two-variable equations are always straight lines, where every point represents a solution. The slope-intercept form y equals m x plus b makes graphing straightforward by identifying the slope and y-intercept. These equations are powerful tools for modeling relationships between two quantities in mathematics, science, and everyday life.