A two-variable linear equation is an equation that can be written in the form A x plus B y equals C, where x and y are the two distinct variables, and A, B, and C are constants or numbers. The important condition is that A and B cannot both be zero. In these equations, the variables are raised only to the power of 1, and there are no products of variables like x times y. When graphed on a coordinate plane, a two-variable linear equation always forms a straight line. For example, the equation 2x plus 3y equals 6 is a two-variable linear equation. We can plot this line by finding points that satisfy the equation, such as the point (0,2) and the point (3,0).
Two-variable linear equations can be written in different forms. The standard form is A x plus B y equals C, which we saw in the previous scene. Another common form is the slope-intercept form, written as y equals m x plus b, where m represents the slope of the line and b is the y-intercept, the point where the line crosses the y-axis. The third important form is the point-slope form, written as y minus y₁ equals m times x minus x₁, where m is the slope and x₁, y₁ represents a specific point on the line. For example, the equation y equals 2x minus 3 is in slope-intercept form. It has a slope of 2, meaning the line rises 2 units for every 1 unit it moves right. The y-intercept is at (0, -3), and we can identify any point on the line, such as (2, 1).
To solve problems involving two-variable linear equations, we often need to work with systems of equations. A system consists of two or more equations that must be satisfied simultaneously. There are several methods to solve these systems. The substitution method involves solving one equation for one variable, then substituting that expression into the other equation. For example, with the system y equals 2x minus 2 and y equals negative x plus 4, we can substitute the first equation into the second to get 2x minus 2 equals negative x plus 4. This gives us 3x equals 6, so x equals 2. Substituting back, we find y equals 2. The elimination method works by adding or subtracting equations to eliminate one variable. Graphically, the solution is the intersection point of the two lines, which in this example is the point (2, 2).
Two-variable linear equations have many real-world applications. They're used in mixture problems, such as combining different solutions or creating alloys with specific properties. They're essential in cost and revenue analysis, helping businesses find break-even points and calculate profits. They also appear in motion problems involving distance, speed, and time relationships, and in work rate problems where we need to determine how long it takes multiple people to complete a task together. Let's look at an example: A coffee shop sells two types of coffee. Type A costs $3 per cup and Type B costs $5 per cup. Yesterday, they sold 100 cups and made $400. How many of each type did they sell? We can set up a system of equations where x represents the number of Type A cups and y represents Type B cups. Our first equation is x plus y equals 100, representing the total number of cups. Our second equation is 3x plus 5y equals 400, representing the total revenue. Solving this system, we find that they sold 50 cups of each type.
To summarize what we've learned about two-variable linear equations: First, a two-variable linear equation has the form A x plus B y equals C, where A and B cannot both be zero. These equations can be written in different forms, including standard form, slope-intercept form, and point-slope form, each useful in different contexts. Systems of linear equations can be solved using various methods such as substitution, elimination, or graphical approaches. When graphed, the solution to a system of two linear equations is the point where the two lines intersect. Finally, these equations have numerous real-world applications, including mixture problems, cost and revenue analysis, motion problems, and work rate calculations. Understanding two-variable linear equations provides a powerful tool for modeling and solving many practical problems.