How do solve the following linear system?: 4y-16=x , 2x+7=-5y 4y16=x,2x+7=5y?

3 Answers
Jun 15, 2017

x = -8.31x=8.31
y = 25/13y=2513

Explanation:

The first equation already has xx as the subject. Therefore you substitute that in place of xx in the second equation.

2(4y - 16) + 7 = -5y -> 8y - 32 + 7 = -5y2(4y16)+7=5y8y32+7=5y

Rearrange the equation to have all unknowns on one side.

13y = 2513y=25

y = 25/13y=2513

Now substitute the yy value into the first equation to get xx.

4(25/13) - 16 = x4(2513)16=x

x = -8.31x=8.31

Jun 15, 2017

(x,y)=color(red)(""(-108/13,25/13))(x,y)=(10813,2513)

Explanation:

Given
[1]color(white)("XXX")4y-16=xXXX4y16=x
[2]color(white)("XXX")2x+7=-5yXXX2x+7=5y

[1]rarr
[3]color(white)("XXX")x=4y-16XXXx=4y16

Using [3] we can replace xx in [2] with 4y-164y16
[4]color(white)("XXX")2(4y-16)+7=-5yXXX2(4y16)+7=5y

Expanding and simplifying the left side of [4]
[5]color(white)("XXX")8y-25=-5yXXX8y25=5y

Adding (5y+25)(5y+25) to both sides of [5]
[6]color(white)("XXX")13y=25XXX13y=25

Dividing both sides of [6] by 1313
[7]color(white)("XXX")y=25/13XXXy=2513

Using [7] we can replace yy in [1] with 25/132513
[8]color(white)("XXX")4 * 25/13-16=xXXX4251316=x

Simplifying [8]
[9]color(white)("XXX")x=-108/13XXXx=10813

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The ugliest part is probably verifying this result by putting the derived values for xx and yy back into the original equations to check that the results are correct.

...with values like these, I would suggest you go to the effort to verify the results (despite the ugliness). Personally I used a spread sheet for this but you might want to use some other method.

Jun 15, 2017

color(red)(y=25/13y=2513 or color(red)(1 12/1311213,color(red)(x=-108/13x=10813 orcolor(red)(-8 4/138413

Explanation:

4y-16=x4y16=x----color(red)((1))(1)

2x+7=-5y2x+7=5y----color(red)((2)(2)

4y-x=164yx=16----color(red)((3)(3)

-5y-2x=75y2x=7----(color(red)(4))(4)

color(red)((3) xx 5(3)×5

20y-5x=8020y5x=80----color(red)((5)(5)

color(red)((4) xx 4(4)×4

-20y-8x=2820y8x=28----color(red)((6)(6)

color(red)((5)+(6)(5)+(6)

-13x=10813x=108

color(red)(x=-108/13x=10813 orcolor(red)(-8 4/138413

Substitute color(red)(x=-108/13x=10813 in (2)

2(color(red)(-108/13))+7=-5y2(10813)+7=5y

-216/13+7=-5y21613+7=5y

Multiply both sides by 13

-65y=-216+9165y=216+91

-65y=-12565y=125

y=cancel125^25/cancel65^13

color(red)(y=25/13 or color(red)(1 12/13

Check:

Substitute color(red)(y=25/13 and color(red)(x=-108/13 in (1)

4(color(red)(25/13))-16=color(red)(-108/13

100/13-16=-108/13

(100-208=-108)/13

Multiply both sides by 65

100-208=-108

color(red)(-108=-108