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Solving Systems of Equations by using the Substitution Method

Objective Learn to solve systems of equations by using the substitution method, and reinforce the geometric concept of what solving the system means.

Graph the systems after you have found the solutions so that the geometric idea that the solution corresponds to the intersection of lines is reinforced.

 

Solving Systems of Equations Algebraically

You should have solved systems of equations graphically by graphing the lines corresponding to the two equations and finding the coordinates of the point of intersection by “inspection”. Often this method is not good enough because it is sometimes difficult to get an exact answer. Instead, there is an algebraic method for solving systems of equations. Let's begin with an example.

 

Example 1

Find the solution to the system of linear equations.

3x + y = 3

2x + 5y = 6

Solution

You can graph the equations to get a rough idea of what the solution set is. The graph of this system of equations is shown below.

Use the graph to estimate the coordinates of the intersection point to be about (0.8, 0.9). So, the solution is approximately x = 0.8, y = 0.9. To find the exact solution, solve the system of equations algebraically.

Key Idea

Use the Addition and Multiplication Properties of Equality to solve systems of equations in two variables, just as equations with one variable are solved.

3x + y = 3  
y = 3 - 3x Subtract 3x from each side.

This equation is solved for y in terms of x . Since the value of y must be the same in both equations, substitute 3 - 3x for y in the second equation.

2x + 5y = 6  
2x + 5(3 - 3x ) = 6 Substitute 3 - 3x for y.
2x + 15 - 15x = 6 Distributive Property
-13x + 15 = 6  

This is an equation involving only one variable, namely x. Now solve the equation for x.

-13x + 15 = 6  
-13x = -9 Subtract 15 from each side.
x Divide each side by -13.
x  

The exact value for the x-coordinate of the solution is . To find the exact value for the y-coordinate, substitute for x in either of the two equations.

Now use the Subtraction Property of Equality to solve for y.

So, the exact solution is given by

This method always works when the system has exactly one solution.

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