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**Solving Systems of Linear Equations in Two Variables**

4.1 Solving Systems of Linear Equations in Two Variables

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**Systems of Linear Equations**

A system of equations consists of two or more equations. The solution of a system of two equations in two variables is an ordered pair (x, y) that makes both equations true.

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Example Determine whether (–3, 1) is a solution of the system. x – y = – 4 2x + 10y = 4 3

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Example Determine whether (4, 2) is a solution of the system. 2x – 5y = – 2 3x + 4y = 4 4

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**Solving Systems of Equations by Graphing**

Since a solution of a system of equations is a solution common to both equations, it is also a point common to the graphs of both equations. To find the solution of a system of two linear equations, we graph the equations and see where the lines intersect. 5

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Example Solve the system of equations by graphing. 6

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Example Solve the system of equations by graphing. 2x – y = 6 x + 3y = 10 7

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Example Solve the system of equations by graphing. –x + 3y = 6 3x – 9y = 9 continued 8

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**Example Solve the system of equations by graphing. x = 3y – 1**

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**Identifying Special Systems of Linear Equations**

There are three possible outcomes when graphing two linear equations in a plane. One point of intersection—one solution Parallel lines—no solution Coincident lines—infinite number of solutions If there is at least one solution, the system is considered to be consistent. If the system defines distinct lines, the equations are independent. 10

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**Possible Solutions of Linear Equations**

Graph Type of System Number of Solutions If the lines intersect, the system of equations has one solution given by the point of intersection. Consistent The equations are independent. (3, 5) Two lines intersect at one point. If the lines are parallel, then the system of equations has no solution because the lines never intersect. Inconsistent The equations are independent. Parallel lines If the lines lie on top of each other, then the system has infinitely many solutions. The solution set is the set of all points on the line. Consistent The equations are dependent. Lines coincide 11

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**The Substitution Method**

Another method that can be used to solve systems of equations is called the substitution method. To use the substitution method, we first need an equation solved for one of its variables. Then substitute that new expression for the variable into the other equation and solve for the other variable. 12

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Example Solve the system using the substitution method. 6x – 4y = 10 Y = 3x - 3 continued 13

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Example Solve the system using the substitution method. 3x – y = 1 4x + y = 6 continued 14

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Example Solve the system using the substitution method. 3x – y = 6 – 4x + 2y = –8 continued 15

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**The Substitution Method**

Solving a System of Two Equations Using the Substitution Method Step 1: Solve one of the equations for one of its variables. Step 2: Substitute the expression for the variable found in Step 1 into the other equation. Step 3: Find the value of one variable by solving the equation from Step 2. Step 4: Find the value of the other variable by substituting the value found in Step 3 into the equation from Step 1. Step 5: Check the ordered pair solution in both original equations. 16

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Example Solve the system: y = 2x – 5 8x – 4y = 20 continued 17

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Example Solve the following system of equations: 3x – y = 4 6x – 2y = 4 continued 18

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**Solving a System Using Elimination**

Another method that can be used to solve systems of equations is called the addition or elimination method. You multiply both equations by numbers that will allow you to combine the two equations and eliminate one of the variables.

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**Example Solve the following system x + y = 7 x – y = 9 x – 5y = -12**

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Example Solve the following system of equations 2x – y = 9 3x + 4y = –14 continued 21

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Example Solve the following system of equations 6x – 3y = –3 4x + 5y = –9 continued 22

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**The Elimination Method**

Solving a System of Two Linear Equations Using the Elimination Method Step 1: Rewrite each equation in standard form, Ax + By = C. Step 2: If necessary, multiply one or both equations by some nonzero number so that the coefficients of a variable are opposites of each other. Step 3: Add the equations. Step 4: Find the value of one variable by solving the equation from Step 3. Step 5: Find the value of the second variable by substituting the value found in Step 4 into either of the original equations. Step 6: Check the proposed solution in both original equations. 23

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**Example Solve the system of equations using the elimination method.**

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