Question

Solve using the method of Substitution
2x-3y=9 and x+4y=2

Answer 1

y = -5/11; x = 42/11

Explanation:

Step 1: Isolate x in x + 4y = 2

• To solve by substitution, we first need to isolate one of the variables in one of the equations.

Let's isolate x in x + 4y = 2:

(x + 4y = 2) - 4y

x = -4y + 2

Step 2: Solve for y by substituting x = -4y + 2 for x in 2x - 3y = 9:

Now we want first solve for y by substituting x = -4y + 2 for x in 2x - 3y = 9:

2(-4y + 2) - 3y = 9

-8y + 4 - 3y = 9

(-8y - 3y) + 4 = 9

(-11y + 4 = 9) - 4

(-11y = 5) / -11

y = -5/11

Thus, y = -5/11.

Step 3: Solve for x by plugging in -5/11 for y in x + 4y = 2:

Now we can solve for by by plugging in -5/11 for y in x + 4y = 2:

x + 4(-5/11) = 2

(x - 20/11 = 2) + 20/11

x = 42/11

Thus, x = 42/11.

Optional Step 4: Check the validity of the answers:

We can check that our answers are correct by plugging in -5/11 for y and 42/11 for x in 2x - 3y = 9 and x + 4y = 2 and seeing if we get the same answers on both sides of the two equations:

Checking y = -5/11 and x = 42/11 in 2x - 3y = 9:

2(42/11) - 3(-5/11) = 9

84/11 + 15/11 = 9

99/11 = 9

9 = 9

Checking y = -5/11 and x = 42/11 in x + 4y = 2:

42/11 + 4(-5/11) = 2

42/11 - 20/11 = 2

22/11 = 2

2 = 2

Thus, our answers are correct.

Answer 2

`\sf  x = (42)/(11)`

`\sf  y = -(5)/(11)`

Explanation:

Given equations:

`\sf 2x - 3y = 9 ......[1]`

`\sf x + 4y = 2........[2]`

Solve equation (2) for x in terms of y:

`\sf x = 2 - 4y`

Substitute the value of x from Step 1 into equation (1):

`\sf 2(2 - 4y) - 3y = 9`

Simplify and solve for y:

`\sf 4 - 8y - 3y = 9`

`\sf -11y = 9 - 4`

`\sf -11y = 5`

`\sf y = -(5)/(11)`

Step 4: Substitute the value of y into equation (2) to find x:

`\sf x + 4(-(5)/(11)) = 2`

`\sf x - (20)/(11 )= 2`

`\sf x = 2 + (20)/(11 )`

`\sf x =( (2*11+ 20))/(11)`

`\sf x = (42)/(11)`

So, the solution to the system of equations is
`\sf  x = (42)/(11)` and
`\sf  y = -(5)/(11)`