Welcome to our lesson on solving systems of two linear equations with two variables. A system of two linear equations with two variables typically looks like this, where a, b, and c are constants. For example, we might have a system like 2x plus 3y equals 8, and 4x minus y equals 5. There are two main methods to solve such systems: substitution method and elimination method. In this lesson, we'll explore both methods and understand how to apply them to find the unique solution where the two lines intersect.
Let's explore the substitution method for solving systems of linear equations. This method involves five key steps. First, choose one equation and solve for one variable in terms of the other. Second, substitute this expression into the other equation. Third, solve the resulting equation to find the value of one variable. Fourth, substitute this value back to find the other variable. Finally, verify your solution in both original equations. Let's apply this to our example system. From the second equation, we can express y as 4x minus 5. Substituting this into the first equation gives us 2x plus 3 times 4x minus 5 equals 8. Simplifying, we get 2x plus 12x minus 15 equals 8, which gives us 14x equals 23. Therefore, x equals 1.64. Substituting back, y equals 4 times 1.64 minus 5, which gives us 1.57. So our solution is the point (1.64, 1.57).
Now let's explore the elimination method for solving systems of linear equations. This method also involves five key steps. First, align the equations so that like terms are in the same column. Second, multiply one or both equations by constants to make the coefficients of one variable either equal or opposite. Third, add or subtract the equations to eliminate one variable. Fourth, solve for the remaining variable. Fifth, substitute back to find the other variable. Let's apply this to our example system. We have 2x plus 3y equals 8, and 4x minus y equals 5. To eliminate y, we can multiply the second equation by 3, giving us 12x minus 3y equals 15. Now the y terms have coefficients 3 and negative 3, so adding the equations will eliminate y. This gives us 14x equals 23, so x equals 1.64. Substituting back into the first equation, we get 2 times 1.64 plus 3y equals 8, which simplifies to 3.28 plus 3y equals 8. Therefore, 3y equals 4.72, and y equals 1.57. Our solution is again the point (1.64, 1.57).
Let's understand the geometric interpretation of systems of linear equations. Each linear equation in two variables represents a straight line in the coordinate plane. The solution to the system is the point of intersection of these lines. In our example, the two lines intersect at the point (1.64, 1.57), which is our solution. However, not all systems have a unique solution. If the lines are parallel, they never intersect, and the system has no solution. This happens when the equations have the same slope but different y-intercepts. If the lines coincide, meaning they're the same line, the system has infinitely many solutions. This occurs when one equation is a multiple of the other. Understanding these geometric interpretations helps us visualize what's happening when we solve systems of equations algebraically.
To summarize what we've learned about solving systems of two linear equations with two variables: A system of two linear equations with two variables consists of two equations, each containing two variables. We explored two main methods for solving such systems. The substitution method involves expressing one variable in terms of the other, then substituting this expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable. Geometrically, each equation represents a line, and the solution is the intersection point of these lines. There are three possible scenarios: a unique solution when the lines intersect, no solution when the lines are parallel, and infinitely many solutions when the lines coincide. These methods are fundamental in algebra and have numerous applications in science, engineering, and economics.