Answer:
- a = -3
- n = 10
- x = 12
- y = 42/13
Explanation:
You want to solve the equations and check the solutions for ...
1. -2a-9=6a+15
2. 14+3n = 5n-6
3. 1/2 x -5=10 - 3/4 x
4. 7/3 y+1=1/6 y+8
Three-step linear equation
Each of these equations has both a variable term and a constant term on each side of the equal sign. The usual 3-step solution is ...
- Subtract one of the variable terms from both sides
- Identify the constant term on the side with 2 terms and subtract that from both sides
- Divide by the coefficient of the variable
This can be made less error-prone by choosing the variable term in Step 1 that has the smallest (least) coefficient. Then the constant in Step 2 will be the constant on the side of the equation that has the variable term with the largest coefficient.
We can do the subtractions of Step 1 and Step 2 both at the same time, reducing the work slightly. Of course, subtraction is the same as adding the opposite. Then what remains is dividing by the coefficient of the variable (or multiplying by its inverse).
1. -2a-9=6a+15
Add 2a -15 to both sides:
-24 = 8a
-3 = a . . . . . . divide by 8
check: -2(-3)-9 = 6(-3) +15 ⇒ -3 = -3 . . . checks OK
2. 14+3n = 5n-6
Add -3n+6 to both sides:
20 = 2n
10 = n . . . . . . divide by 2
check: 14 +3(10) = 5(10) -6 ⇒ 44 = 44 . . . checks OK
3. 1/2 x -5=10 - 3/4 x
Add 3/4x +5 to both sides:
5/4x = 15
x = 12 . . . . . . multiply by 4/5
check: 1/2(12) -5 = 10 -3/4(12) ⇒ 1 = 1 . . . checks OK
4. 7/3 y+1=1/6 y+8
Add -1/6y -1 to both sides:
13/6y = 7
y = 42/13 . . . . . multiply by 6/13
check: (7/3)(42/13) +1 = (1/6)(42/13) +8 ⇒ 98/13 +1 = 7/13 +8
⇒ 8 7/13 = 8 7/13 . . . checks OK