Simultaneous Equations

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#14095

Solve simultaneously: 2x + y = 8, y = 4x – 10.

substitute y = 4x – 10 into second equation
2x + (4x – 10) = 8
6x – 10 = 8
6x = 18
x = 3

y = 4(3) – 10
y = 12 – 10
y = 2
∴ x = 3, y = 2

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#14096

Solve simultaneously: 7a + 3b = 4, 5a + 2b = 3.

7a + 3b = 4    ×2
5a + 2b = 3    ×3

14a + 6b = 8    –
15a + 6b = 9
-a = -1
a = 1

7(1) + 3b = 4
7 + 3b = 4
3b = -3
b = -1
∴ a = 1, b = -1

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#14097

Solve 2x + y = 4 and 5x – 2y = 9.	Sometimes neither method is ready to go, so we have to do extra work.
(2x + y = 4) × 2 →   4x + 2y = 8 4x + 2y = 8 5x – 2y = 9 9x = 17 x = 18/9 2 × 18/9 + y = 4 37/9 + y = 4 y = 2/9 ∴ x = 18/9, y = 2/9	to use the elimination method when none of the x’s or y’s match, multiply the expressions to make them match. if we multiply the first equation by 2, we will get 4x + 2y = 8, now the y’s are both 2, so we can eliminate by adding them since 2y + -2y = 0 4x + 5x = 9x and 8 + 9 = 17 solve for x and then substituted the x value back into any of the original equations to find y
Solve 2x + y = 4 and 5x -2y = 9.	Using the substitution method can work as well.
y = 4 – 2x	rewrite the first equation so that it says y =
5x – 2(4 – 2x) = 9 5x – 8 + 4x = 9 9x – 8 = 9 9x = 17 x = 18/9	substitute y = 4 – 2x into the second equation (remember be super careful of the minus signs) solve for x
y = 4 – 2 × 18/9 y = 2/9	substitute the x value into the new equation you made that says y =  solve for y
∴ x =  18/9  y = 2/9

 

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#14502

Solve: 3x – 5y = 28 and x + 4y = 15

multiply the second one by 3 to make the x’s the same

3x + 12y = 45  –
3x – 5y = 28

3x – 3x + 12y – -5y = 45 – 28

17y = 17

y = 1

x + 4(1) = 15

x + 4 = 15

x = 11

∴ x = 11, y = 1

[Admin]

#18925

Solve 2x + 3y = 4 and 2x – 4y = -10.
2x + 3y = 4 2x – 4y = -10 7y = 14 y = 2	Elimination Method this method works by adding or subtracting the equations to eliminate one of the letters and is the best method when the x‘s or y‘s are already the same since both the x’s are 2x, if we subtract them 2x – 2x = 0, so the x’s are eliminated, leaving us only with y to solve subtract: 2x – 2x = 0 3y – (-4y) = 7y    just be extra careful with the minus minus – which makes a plus solve for y
2x + 6 = 4 2x = -2 x = -1	once we have a value for y, we can substitute it back into one of the original equations (it doesn’t matter which one) to the solve for x
22x = 220	add 200 to both sides
x = 10	divide both sides by 22

 

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#18926

Solve 2x – 4y = 2 and x = -2y + 9	Substitution Method this is the best method when one of the equations already say x = or y =
2(-2y + 9) – 4 = 2 -4y + 18 – 4y = 2 -8y = -16 y = 2	in this instance the second equation is x = -2y + 9 so we replace the x in the first equation with the -2y + 9 and then solve for y
x = -2(2) + 9 x = -4 + 9 x = 5	once you have solved for y, substitute into one of the first equations (try and pick the one that is the easiest) and solve for x
∴ x = 5, y = 2

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