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# Module 5 - Linear Eq

### Math 113 Online: Module 5 - Linear Equations and Substitution Method

Question | Answer |
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Determine whether each ordered pair is a solution of the system of linear equations. {x+y=8 and 3x+4y=29 a) (2,6) b) (3,5) | Plug in each ordered pair into both equations to determine if said ordered pairs are solutions to the linear equations. 2+6=9; 3(2)+4(6)=29/6+24=30 and 3+5=8; 3(3)+4(5)=29/9+20=29. Ordered pair a isn't a solution, but pair b is. |

Solve the System of linear equations by graphing. {x+y=9 and x-y=5 | Once you've graphed both equations, find the point at which both lines intersect. The ordered pair should satisfy the equations. The solution of the system is (7,2) because it completes both linear equations. 7+2=9;7-2=5 |

Without graphing, decide. a) Are the graphs of the equations identical lines, parallel lines, or lines intersecting at a single point? and b) How many solutions does the system have? {9x+7=27 and x+8y=8 | Rewrite both equations in slope intercept form y=-9+27 and y=-1/8x+1. Identify the slopes and y-intercepts for both lines. The slopes for the lines are -9 and -1/8. The slopes are different, and must intersect at one point; meaning one solution. |

Solve the system of equations using the substitution method. {x+y=10 and x=4y | The first step is to solve one of the equations for one variable. The second equation; x=4y has already been solved for x, so we'll use it. 4y+y=10 (Combine like terms) 5y=10; y=2 x=4(2); x=8. So the solution is (8,2) |

Solve the system of equations by the substitution method. {y=5x+7 and y=7x+8 | Substitute 5x+7 for y in the 2 equation. 5x+7=7x+8 isolate the x terms on the left side by subtracting 7x and adding 7. -2x=1; x=-1/2 Plug in x for the 1st equation. y=5(-1/2)+7; y=-5/2+7. Simplify. y=9/2 So the ordered pair is (-1/2,9/2) |

Solve the system of equations by the substitution method. {12x+3y=8 and -4x=y+8 | Neither equation is solved for x or y, so we'll isolate y on the right side of equation 2. -4x-8=y Now fill in the 1st equation 12x+3(-4x-8)=8 Use the distributive property on the left side. 12x-12x-24=8 -24=8 is a false statement/no solution. |

Solve the system of equations by the substitution method. {4x-y=3 and 5x-2y=12 | Neither is solved, so we'll isolate y in the 1st equation. -y=3-4x Solve for y y=4x-3 Now substitute y in the 2nd equation. 5x-2(4x-3)=12 Distributive property. 5x-8x+6=12; -3x+6=12, isolate x. -3x=6, solve for x, x=-2. Replace x in the first equation. |

Solve the system of equations by the substitution method. {4x-y=3 and 5x-2y=12 (Continued) | y=4(-2)-3, y=-8-3, y=-11. So the ordered pair is (-2,-11) |

Solve the system of equations by the substitution method. {3x+12y=15 and 4x+18y=22 | Neither is solved, so we'll isolate x in the 1st equation. 3x=15-12y, solve for x. x=5-4y, now substitute x in the 2nd equation. 4(5-4y)+18y=22 Distribute. 20-16y+18y=22, combine like terms. 20+2y=22 isolate y on the left side. 2y=2; y=1 |

Solve the system of equations by the substitution method. {3x+12y=15 and 4x+18y=22 (Continued) | Replace y in the first equation. x=5-4(1) Solve. x=1 So the ordered pair is (1,1) |