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System of Linear Equations

System of Linear Equations. The whole purpose of education is to turn mirrors into windows ~Sydney J. Harris. What is a System of Linear Equations?. A system of linear equations is simply two or more linear equations using the same variables.

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System of Linear Equations

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  1. System of Linear Equations

  2. The whole purpose of education is to turn mirrors into windows ~Sydney J. Harris

  3. What is a System of Linear Equations? • A system of linear equations is simply two or more linear equations using the same variables. • We will only be dealing with systems of two equations using two variables, x and y. • If the system of linear equations is going to have a solution, then the solution will be an ordered pair (x , y) where x and y make both equations true at the same time. • We will be working with the graphs of linear systems and how to find their solutions graphically.

  4. y x (1 , 2) Solve Linear Systems Consider the following system: x – y = –1 x + 2y = 5 Using the graph to the right, we can see that any of these ordered pairs will make the first equation true since they lie on the line. We can also see that any of these points will make the second equation true. However, there is ONE coordinate that makes both true at the same time… The solution is simply the point of intersection

  5. Solution to Sample We must always verify a proposed solution algebraically. We propose (1,2) as a solution, so now we plug it in to both equations to see if it works: x = 1 y = 2 x – y = -1 (1) – 2 = - 1 = -1 x + 2y = 5 (1) + 2(2) = 1 + 4 = 5 = 5   Yes, (1,2) Satisfies both equations!

  6. Solving System of Linear Equations There are several methods of solving systems of linear equations. Each is best used in different situations. • Graphing Method • Substitution Method • Elimination (Addition) Method

  7. Graphing Method To solve a system of linear equations by the graphing method, there are three basic steps to follow: • Graph the equations on the same coordinate plane • Use the slope and y–intercept if needed. Be sure to use a ruler and graph paper! • Estimate where the graphs intersect. • This is the solution! LABEL the solution! • Check to make sure your solution makes both equations true.

  8. y x (3 , 1) Label the solution! Graphing Method Solve the following system of equations by graphing. 3x + 6y = 15 –2x + 3y = –3 Step 1: Graph both equations on the same coordinate plane Step 2: Estimate where the graphs intersect. LABEL the solution! Step 3: Check to make sure your solution makes both equations true. 3x + 6y = 15 3(3) + 6(1) = 9 + 6 = 15 -2x + 3y = -3 -2(3) + 3(1) = -6 + 3 = -3 -2x + 3y = -3 3y = 2x – 3 y = 2/3x – 1 3x + 6y = 15 6y = -3x + 15 y = -1/2 x + 5/2

  9. Types of Solutions • If the lines cross once, there will be one solution. • If the lines are parallel, there will be no solutions. • If the lines are the same, there will be an infinite number of solutions.

  10. Substitution Method Substitution method is used when it appears easy to solve for one variable in terms of the other. The goal is to reduce the system to two equations of one unknown each. Consider the following: 2x + 4y = 28 y = 3x

  11. Substitution Method Solve using substitution. y = 3x 2x + 4y = 28 y = 3x 2x + 4(3x) = 28 y = 3(2) 2x + 12x = 28 y = 6 14x = 28 x = 2 (2,6)

  12. Substitution Method • To solve a system of equations by substitution… • 1. Solve one equation for one of the variables. • 2. Substitute the value of the variable into the other equation. • Simplify and solve the equation. • Substitute back into either equation to find the value of the other variable. • Check the solution

  13. Solve using substitution. -3x + y = -17 3x + 2y = 2 y = 3x – 17 3x + 2(3x – 17) = 2 3x + 6x – 34 = 2 Step 1: Solve one equation for one of the variables 9x – 34 = 2 9x = 36 Step 2: Substitute the value of the variable into the other equation. x = 4 -3x + y = -17 Step 3: Simplify and solve the equation. -3(4) + y = -17 Step 4: Substitute back into either equation to find the value of the other variable. -12 + y = -17 (4,-5) y = -5

  14. Check Solution We must always verify a proposed solution algebraically. We propose (4,-5) as a solution, so now we plug it in to both equations to see if it works: -3x + y = -17 3x + 2y = 2 -3x + y = -17 -3(4) + (-5) = -12 – 5 = -17 = -17 3x + 2y = 2 3(4) + 2(-5) = 12 + -10 = 2 = 2   Yes, (4,-5) Satisfies both equations!

  15. Elimination Method Elimination method is used when it appears easy to eliminate one variable from the system by adding the two equations together The elimination method makes use of the addition principle of equality if a = b, then a + c = b + c

  16. Elimination Method Solve using the elimination method. x – y = 7 x + y = 3 x + y = 3 5 + y = 3 2x + 0y = 10 y = -2 2x = 10 x = 5 (5,-2)

  17. Elimination Method • To solve a system of equations by elimination… • 1.Put equations in standard form (Ax + By = C) • 2. Determine which variable to eliminate. • Add or subtract the equations and solve for the variable. • Substitute back into either equation to find the value of the other variable. • Check the solution

  18. Solve using the elimination method. 2x + 3y = 11 -2x + 9y = 1 2x + 3y = 11 2x + 3(1) = 11 0x + 12y = 12 2x + 3 = 11 12y = 12 2x = 8 y = 1 x = 4 (4,1)

  19. Elimination Method Solve using the elimination method. 2x + 3y = 11 -2x + 9y = 1 2x + 3y = 11 2x + 3(1) = 11 0x +12y = 12 2x + 3 = 11 2x = 8 12y = 12 (4,1) y = 1 x = 4 Step 1: Put in standard form Step 2: Determine which variable to eliminate Step 3: Add equations and solve Step 4: Substitute and solve for other variable Done Variable x

  20. Check Solution We must always verify a proposed solution algebraically. We propose (4,1) as a solution, so now we plug it in to both equations to see if it works: 2x + 3y = 11 -2x + 9y = 1 2x + 3y = 11 2(4) + 3(1) = 8 + 3 = 11 = 11 -2x + 9y = 1 -2(4) + 9(1) = -8 + 9 = 1 = 1   Yes, (4,1) Satisfies both equations!

  21. Elimination Method Sometimes you may need to utilize the multiplication property of equality if a = b, then ac = bc to help eliminate a variable

  22. Solve using the addition method. 3x – y = 8 x + 2y = 5 write in standard form multiply as needed (2)( ) ( )(2) 3x – y = 8 add the equations x + 2y = 5 substitute 6x – 2y = 16 x + 2y = 5 x + 2y = 5 (3) + 2y = 5 7x = 21 2y = 2 x = 3 y = 1 (3,1)

  23. Solve using the addition method. 3x + 5y = 12 4x – 3y = -13 write in standard form multiply as needed (eliminate x variable) (3)( ) ( )(3) 3x + 5y = 12 (5)( ) ( )(5) 4x – 3y = -13 add the equations substitute 9x + 15y = 36 3x + 5y = 12 20x – 15y = -65 3(-1) + 5y = 12 29x = -29 5y = 15 x = -1 y = 3 (-1,3)

  24. Parallel Lines (no solution) y = -2x – 3 y = -2x + 5 write in standard form (-1)( ) ( )(-1) multiply as needed 2x + y = -3 add the equations 2x + y = 5 -2x – y = 3 This is a contradiction since 0 does not equal to 8 2x + y = 5 0 + 0 = 8 0 = 8 No solution

  25. Same Lines (infinite solutions) 6x + 2y = 4 y = -3x + 2 write in standard form multiply as needed 6x + 2y = 4 add the equations (-2)( ) ( )(-2) 3x + y = 2 6x + 2y = 4 This is a always a true statement regardless of values for x and y -6x – 2y = -4 0 + 0 = 0 0 = 0 Infinite solutions

  26. The sum of a number and twice another number is 13. The first number is 4 larger than the second number. What are the numbers? Let x = the first number Let y = the second number x = y + 4 x + 2y = 13 y + 4 + 2y = 13 3y + 4 = 13 x = 3 + 4 x = y + 4 x = 7 3y = 9 y = 3 Use substitution

  27. The admission fee at a small fair is $1.00 for children and $4.00 for adults. On a certain day, 1,000 people entered the fair and $2,200 is collected. How many children and how many adults attended? Let x = number of adults Let y = number of children (-1)( ) ( )(-1) x + y = 1000 4x + y = 2200 Use elimination -x – y = -1000 4x + y = 2200 x + y = 1000 400 + y = 1000 3x = 1200 y = 600 children x = 400 adults

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