A linear equation in two variables is an equation of the form ax plus by equals c, where x and y are variables, and a, b, c are constants. When we have two such equations, they form a system. The solution is the point where both lines intersect on the coordinate plane.
The substitution method involves four steps. First, solve one equation for one variable. From x plus y equals 5, we get y equals 5 minus x. Second, substitute this expression into the other equation: 2x minus the quantity 5 minus x equals 1. Third, solve for x: 3x equals 6, so x equals 2. Finally, substitute back to find y equals 3. The solution is the point 2, 3.
The elimination method involves making coefficients equal and then eliminating variables. Starting with 3x plus 2y equals 12 and x minus y equals 1, we multiply the second equation by 2 to get 2x minus 2y equals 2. Adding the equations eliminates y: 5x equals 14, so x equals 14 fifths. Substituting back gives y equals 9 fifths. The solution is 14 fifths, 9 fifths.
A system of linear equations can have three types of solutions. One solution occurs when the lines intersect at exactly one point. No solution occurs when the lines are parallel and never meet. Infinite solutions occur when the equations represent the same line, so every point on the line is a solution.
Linear equations in two variables have many real-world applications including business cost analysis, physics motion problems, economics supply and demand, and engineering optimization. We learned two main solution methods: substitution and elimination. The substitution method involves solving for one variable first, while the elimination method makes coefficients equal. Both methods find the intersection point of two lines, which represents the solution to the system.