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Systems of Linear Equations <ul><li>4-1 Systems of Linear Equations in Two Variables </li></ul>

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4-1 Systems of Linear Equations in Two Variables <ul><li>Deciding whether an ordered pair is a solution of a linear system. </li></ul><ul><li>The solution set of a linear system of equations contains all ordered pairs that satisfy all the equations at the same time. </li></ul><ul><li>Example 1: Is the ordered pair a solution of the given system? </li></ul><ul><li>2x + y = -6 Substitute the ordered pair into each equation. </li></ul><ul><li> x + 3y = 2 Both equations must be satisfied. </li></ul><ul><li>A) (-4, 2) B) (3, -12)‏ </li></ul><ul><li>2(-4) + 2 = -6 2(3) + (-12) = -6 </li></ul><ul><li>(-4) + 3(2) = 2 (3) + 3(-12) = 2 </li></ul><ul><li>-6 = -6 -6 = -6 2 = 2 -33  -6 </li></ul><ul><li> Yes  No </li></ul>

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4-1 Systems of Linear Equations in Two Variables <ul><li>Solving Linear Systems by Graphing. </li></ul><ul><li>One way to find the solution set of a linear system of equations is to graph each equation and find the point where the graphs intersect. </li></ul><ul><li>Example 1: Solve the system of equations by graphing. </li></ul><ul><li>A) x + y = 5 B) 2x + y = -5 </li></ul><ul><li> 2x - y = 4 -x + 3y = 6 </li></ul><ul><li>Solution: {(3,2)} Solution: {(-3,1)} </li></ul>

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4-1 Systems of Linear Equations in Two Variables <ul><li>Solving Linear Systems by Graphing. </li></ul><ul><li>There are three possible solutions to a system of linear equations in two variables that have been graphed: </li></ul><ul><li>1) The two graphs intersect at a single point. The coordinates give the solution of the system. In this case, the solution is “consistent” and the equations are “independent” . </li></ul><ul><li>2) The graphs are parallel lines. (Slopes are equal) In this case the system is “inconsistent” and the solution set is 0 or null. </li></ul><ul><li>3) The graphs are the same line. (Slopes and y-intercepts are the same) In this case, the equations are “dependent” and the solution set is an infinite set of ordered pairs. </li></ul>

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4-1 Systems of Linear Equations in Two Variables <ul><li>Solving Linear Systems of two variables by Method of Elimination. </li></ul><ul><li>Remember: If a=b and c=d, then a + c = b + d. </li></ul><ul><li>Step 1: Write both equations in standard form </li></ul><ul><li>Step 2: Make the coefficients of one pair of variable terms opposite </li></ul><ul><li>(Multiply one or both equations by appropriate numbers so that the sum of the coefficients of either x or y will be zero.)‏ </li></ul><ul><li>Step 3: Add the new equations to eliminate a variable </li></ul><ul><li>Step 4: Solve the equation formed in step 3 </li></ul><ul><li>Step 5: Substitute the result of Step 4 into either of the original equations and solve for the other value. </li></ul><ul><li>Step 6: Check the solution and write the solution set. </li></ul>

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4-1 Systems of Linear Equations in Two Variables <ul><li>Solving Linear Systems of two variables by Method of Elimination. </li></ul><ul><li>Example 2: Solve the system : 2x + 3y = 19 </li></ul><ul><li>Step 1: Both equations are in standard form 3x - 7y = -6 </li></ul><ul><li>Step 2: Choose the variable x to eliminate: Multiply the top equation by 3, the bottom equation by -2 </li></ul><ul><li>3[2x + 3y = 19] 6x + 9y = 57 </li></ul><ul><li> -2[3x - 7y = -6] -6x +14y = 12 </li></ul><ul><li>Step 3: Add the new equations to eliminate a variable </li></ul><ul><li> 0x + 23y = 69 </li></ul><ul><li>Step 4: Solve the equation formed in step y = 3 </li></ul><ul><li>Step 5: Substitute the result of Step 4 into either of the original equations and solve for the other value. 2x + 3(3) = 19 </li></ul><ul><li>2x = 10 </li></ul><ul><li> x = 5 Solution Set: {(5,3)} </li></ul><ul><li>Step 6: Check the solution and write the solution set. </li></ul>

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4-1 Systems of Linear Equations in Two Variables <ul><li>Solving Linear Systems of two variables by Method of Elimination. </li></ul><ul><li>Example 3: </li></ul><ul><li>Solve the system : </li></ul><ul><li>2[2x - 3y = 1] 4x - 6y = 2 </li></ul><ul><li> -3[3x - 2y = 9] -9x + 6y = -27 </li></ul><ul><li>-5x + 0y = -25 </li></ul><ul><li> x = 5 3(5) - 2y = 9 </li></ul><ul><li> -2y = -6 </li></ul><ul><li>Solution Set: {(5,3)} y = 3 </li></ul>

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4-1 Systems of Linear Equations in Two Variables <ul><li>Solving Linear Systems of two variables by Method of Elimination. </li></ul><ul><li>Example 4: </li></ul><ul><li>Solve the system : 2x + y = 6 </li></ul><ul><li> -8x - 4y = -24 </li></ul><ul><li>4[2x + y = 6] 8x + 4y = 24 </li></ul><ul><li> -8x -4y = -24 -8x - 4y = -24 </li></ul><ul><li> 0 = 0 True </li></ul><ul><li>Solution Set: {(x,y)| 2x + y = 6} </li></ul><ul><li>Note: When a system has dependent equations and an infinite number of solutions, either equation can be used to produce the solution set. Answer is given in set-builder notation. </li></ul>

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4-1 Systems of Linear Equations in Two Variables <ul><li>Solving Linear Systems of two variables by Method of Elimination. </li></ul><ul><li>Example 5: </li></ul><ul><li>Solve the system : 4x - 3y = 8 </li></ul><ul><li> 8x - 6y = 14 </li></ul><ul><li>-2[4x - 3y = 8] -8x + 6y = -16 </li></ul><ul><li> 8x - 6y = 14 8x - 6y = 24 </li></ul><ul><li> 0 = 8 False </li></ul><ul><li>Solution Set: 0 or null </li></ul><ul><li>Note: There are no ordered pairs that satisfy both equations. The lines are parallel. There is no solution. </li></ul>

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4-1 Systems of Linear Equations in Two Variables <ul><li>Solving Linear Systems of two variables by Method of Substitution. </li></ul><ul><li>Step 1: Solve one of the equations for either variable </li></ul><ul><li>Step 2: Substitute for that variable in the other equation </li></ul><ul><li>(The result should be an equation with just one variable)‏ </li></ul><ul><li>Step 3: Solve the equation from step 2 </li></ul><ul><li>Step 4: Substitute the result of Step 3 into either of the original equations and solve for the other value. </li></ul><ul><li>Step 6: Check the solution and write the solution set. </li></ul>

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4-1 Systems of Linear Equations in Two Variables <ul><li>Solving Linear Systems of two variables by Method of Substitution. </li></ul><ul><li>Example 6: Solve the system : 4x + y = 5 </li></ul><ul><li>2x - 3y =13 </li></ul><ul><li>Step 1: Choose the variable y to solve for in the top equation: </li></ul><ul><li>y = -4x + 5 </li></ul><ul><li>Step 2: Substitute this variable into the bottom equation </li></ul><ul><li> 2x - 3(-4x + 5) = 13 2x + 12x - 15 = 13 </li></ul><ul><li>Step 3: Solve the equation formed in step 2 </li></ul><ul><li> 14x = 28 x = 2 </li></ul><ul><li>Step 4: Substitute the result of Step 3 into either of the original equations and solve for the other value. 4(2) + y = 5 </li></ul><ul><li>y = -3 </li></ul><ul><li>Solution Set: {(2,-3)} </li></ul><ul><li>Step 5: Check the solution and write the solution set. </li></ul>

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4-1 Systems of Linear Equations in Two Variables <ul><li>Solving Linear Systems of two variables by Method of Substitution. </li></ul><ul><li>Example 7: </li></ul><ul><li>Solve the system : </li></ul><ul><li> y = -2x + 2 </li></ul><ul><li> -2x + 5(-2x + 2) = 22 -2x - 10x + 10 = 22 </li></ul><ul><li>-12x = 12 </li></ul><ul><li> x = -1 2(-1) + y = 2 </li></ul><ul><li> y = 4 </li></ul><ul><li>Solution Set: {(-1,4)} </li></ul>