6.2 Using Substitution to Solve Systems

Example: Making a Substitution for y Solve the system Equation (1) Equation (2)

Solution From equation (1). We know that for any solution of the system, the value of y is equal to the value of x + 3. So, we substitute x + 3 for y in equation (2): Equation (2)

Solution Note that, by making this substitution, we now have an equation in one variable. Next, we solve that equation for x:

Solution Thus the x-coordinate of the solution is 1. To find the y-coordinate, we substitute 1 for x in either of the original equations and solve for y: Equation (1)

Solution So, the solution is (1, 4). We can check that (1, 4) satisfies both of the system’s equations:

Solution Or, we can verify that (1, 4) is the solution by graphing equations (1) and (2) and checking that (1, 4) is the intersection point of the two lines. To do so on a graphing calculator, we must first solve equation (2) for y:

Solution

Using Substitution to Solve a Linear System To use substitution to solve a system of two linear equations, 1. Isolate a variable on one side of either equation. 2. Substitute the expression for the variable found in step 1 into the other equation. 3. Solve the equation in one variable found in step 2. 4. Substitute the solution found in step 3 into one of the original equations, and solve for the other variable.

Example: Isolating a Variable and Then Using Substitution Solve the system Equation (1) Equation (2)

Solution We begin by solving for one of the variables in one of the equations. We can avoid fractions by choosing to solve equation (1) for y: Equation (1)

Solution Next, we substitute –3x – 7 for y in equation (2) and solve for x: Equation (2)

Solution Finally, we substitute –2 for x in the equation y = –3x – 7 and solve for y: The solution is (–2, –1). We could then verify our work by checking that (–2, –1) satisfies both of the original equations.

Example: Applying Substitution to an Inconsistent System Consider the linear system Equation (1) Equation (2) The graphs of the equations are parallel lines (why?), so the system is inconsistent and the solution is the empty set. What happens when we solve this system by substitution?

Solution We substitute 3x + 2 for y in equation (2) and solve for x: false We get the false statement 2 = 4.

Inconsistent System of Equations If the result of applying substitution to a system of equations is a false statement, then the system is inconsistent; that is, the solution set is the empty set.

Example: Applying Substitution to an Dependent System In an example in Section 6.1, we found that the system Equation (1) Equation (2) is dependent and that the solution set is the infinite set of solutions of the equation y = 3x – 5. What happens when we solve this system by substitution?

Solution We substitute 3x – 5 for y in equation (2) and solve for x: true We get the true statement 10 = 10.

Dependent System of Equations If the result of applying substitution to a system of equations is a true statement that can be put into the form a = a, then the system is dependent; that is, the solution set is the set of ordered pairs represented by every point on the (same) line.