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4x + 13y = 40 4x + 3y = -40

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4 Answers

Solving Systems of Linear Equations
         2 Ways.
               1. Substitution
               2. Linear Combination/ Elimination Method

Problem:
1. 4x + 13y = 40
2. 4x + 3y= -40

USING.                 Linear Combination/ Elimination Method

STEPS

1. Label each equation with number 1- however many there are

2. Choose a VARIABLE to cancel when you ADD the linear equations
(easiest one...ex. Whole number if possible)


3. Multiply one of the equations by the opposite(-1 in this case) 1. 4x + 13y = 40
                                                                                                     2. (-1)( 4x + 3y) =( -40)(-1)
                         or whatever number will cancel the variable
4. Add both equations together to cancel one of the variables ( x in this case).


5. Solve for other variable(y in this case).                                      1.   4x + 13y = 40
                                                                                                       2. - 4x +- 3y = 40
                                                                                                                      10y=80
                                                                                                                          y=8
6. Plug in value for solved variable into one of the ORIGINAL equations. y=8. 1. 4x + 13y = 40
                                                            1. 4x + 13(8)= 40
                                                            1. 4x +104= 40
                                                                4x=40+-104=-64
                                                                  x=-64/4=-16


7. Check solution for x and y in both original equations by plugging in your answers to x and y to see if the equations are true(equal on both side)
                                                  x=-16, y=8. Substitute  into Equation 2. 4x + 3y= -40
                                                                                                                     -64 + 24=-40   Check
Therefore, these lines intersect at point ( -16, 8).

NOTE:    If the point would not have checked then here are the possible solutions to linear equations regardless of how you solve them.

POSSIBLE SOLUTIONS
1. There is a solution that makes both equations true so they INTERSECT at that point. Solution is the point that is a solution for all linear equations (x,y).
2. There is NOT a solution to the equations they DO NOT INTERSECT And are PARALLEL. Solution is { }
3. All real numbers make both equations true so they are COINCIDENT LINES aka the same line. Solution is all real numbers or which ever number family you are working with.

Comments

D...I hope the previous response helps with the answer. Mind if I had a tiny hint so that problems like this won't be impossible to answer....to your problem 4x + 13y = 40 4x + 3y = -40 When you saw the problem, what was the problem asking you to do? Find the value of x and the value of y, correct? That should by your first question...what will the answer look like. its going to be x = some value and y = some value If you're able to see this...then you'll be able to begin to answer the problems. As the previous response demonstrates, the steps it takes to solve for the variables. If you're confused...in the dark about what the problem is requiring you to do, It will be difficult to begin to answer the problem. If this is the case, discuss this with your instructor/teacher.
Use a step by step approach and it helps to number the lines. 1) Given:4x + 13y = 40 2) Given:4x + 3y = -40 3) Modify one of the equations to express one of the variables (x or y) in terms of the other. Line 1) can be re-arranged to the form below. 4) x=10-13/4y 5) Substitute the expression for x into line 2) resulting in the below. 6) 40-13y+3y=-40 7) Solve 6) for y 8) Simplify by subtracting 40 from both sides 9) -13y+3y=-40-40=-80 10) combine the left side into -10y 11) -10y=-80 12)divide both sides by -10 13) y=8 14) substitute 8 for y in either 1) or 2). 15) Choosing 1) 4x + 13(8)=40 16) Solving for x results in that x=-64/4=-16 Summary is that each equation has the same result when x=-16 & y=8.

We need to eliminate one of the variables.  Here, the easiest way to do it is to subtract one equation from the other.

(4x + 13y = 40) - (4x + 3y = -40) = (10y = 80).  The x term drops out.

10y = 80, so y = 8,  Now substitute 8 for y in both equations and confirm that both produce the same value for x.

4x + (13 * 8) = 40

4x + 104 = 40

4x = -64

x = -16

4x + (3 * 8) = -40

4x + 24 = -40

4x = -64

x = -16

 

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