Intermediate Algebra Section 4.1 Systems of Linear Equations in Two Variables

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1 Intermediate Algebra Section 4.1 Systems of Linear Equations in Two Variables A system of equations involves more than one variable and more than one equation. In this section we will focus on systems containing two equations and two variables. For example, a system of two equations in x and y is x y = 4 x + 2y = 5 A solution is an ordered pair ( x, y) that satisfies both equations. Since a solution of a system of two equations in two variables is a solution common to both equations, it is also a point common to the graphs of both equations. This is the point where the two graphs intersect. A system of equations that has at least one solution is said to be a consistent system. A system that has no solution is said to be inconsistent. A system that has infinitely many solutions is said to be coincident.

2 Section 4.1 Systems of Linear Equations in Two Variables page 2 Example: Determine whether the ordered pair (4, 7) is a solution x + y = 11 to the system:. 2x 5y = 43 Solving a system of linear equations by graphing is not usually a very effective way to solve systems. There are other methods that should be used because they are accurate. We will look at a couple of other methods for solving linear equations. In this section, we will solve systems using the substitution method. Solving a System of Linear Equations by the Substitution Method. 1. Solve one of the equations for one of its variables. 2. Substitute the expression for the variable found in step 1 into the other equation. 3. Solve the equation from step 2 to find the value of one variable. 4. Substitute the value found in step 3 in any equation containing both variables to find the value of the other variable. 5. Check the proposed solution in the original equation.

3 Section 4.1 Systems of Linear Equations in Two Variables page 3 Example: Solve the following systems first by graphing and then y using substitution. a) y = 2x + 3 5y 7x = x y b) x + y = 3 2x + 2y = x

4 Section 4.1 Systems of Linear Equations in Two Variables page 4 A consistent system of linear equations has at least one solution, and an inconsistent system has no solution. You can see from the graph in the previous example that an inconsistent system consists of lines that have the same slope and different y -intercept. Algebraically, you will know that you have an inconsistent system if the variables are all eliminated and you are left with a false statement, as was the case in the previous example. There are two types of consistent linear systems. Linear systems whose graphs intercept at exactly one point, which means the lines have different slopes, and linear systems that intersect at infinitely many points, which means the lines have the same slope and the same y -intercept. Another method we can use to solve systems of linear equations is called the elimination method. The elimination method is based on the addition property of equality, adding equal quantities to both sides of an equation does not change the solution of the equation. When using the elimination method, we want to make it so that the coefficient of one of the variables are opposites, so that when we add the equations one of the variables is eliminated. Solving a System of Two Linear Equations by the Elimination Method 1. Rewrite each equation in standard form Ax + By = C. 2. If necessary, multiply one or both equations by a nonzero number so that the coefficients of a chosen variable in the system are opposites. 3. Add the equations. 4. Find the value of one variable by solving the resulting equation from step Find the value of the second variable by substituting the value found in step 4 into either of the original equations. 6. Check the proposed solution in the original equation.

5 Section 4.1 Systems of Linear Equations in Two Variables page 5 Example: a) x + 3y = 2 x + 2y = 3 Solve the following systems using the elimination method. b) 3x + 4 y = 2 2x + 5y = 1

6 Section 4.1 Systems of Linear Equations in Two Variables page 6 c) x + 3y = 6 3x 9y = 9

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