2a-b equals 17 3a plus 4b equals -13?

Answer 1

2a-b equals 17 3a plus 4b equals -13?

Elimination Method

If you can multiply one of the equations by a number and eliminate one of the variables, that is an easy to find the value of the other variable. I see -b in Eq. #1 and +4b in Eq. #2. I can multiply by Eq. #1 by 4 and add to Eq. #2 eliminate b.

Eq. #1 = 2a - b = 17

Eq. #2 = 3a + 4b = -13

4* (2a - b = 17) = 8a - 4b = 68 Now add 4* Eq. #1 to Eq. #2

..8a - 4b = 68

+.3a + 4b = -13

.11a + 0b = 55

11a = 55 divide both sides by 11

a = 5

Substituting a = 5 into Eq. #1

(2 * 5) - b = 17

10 - b = 17 subtract 10 from both sides

- b = 17 - 10

-b = 7 Divide by -1

b = -7

Check in Eq. #2, by substituting a = 5 and b = -7

3a + 4b = -13

(3 * 5) + (4 * -7) = -13

15 + -28 = -13

-13 = -13

The other way of solving simultaneous equations is Substitution.

Solve for one variable in terms of the other variable.

Eq. #1 = 2a - b = 17

Eq. #2 = 3a + 4b = -13

Solve Eq.#1 for a in terms of b.

2a - b = 17 add +b to both sides

2a = b + 17 divide (b + 17) by 2

a = (b + 17) ÷ 2

Substitute a = (b + 17) ÷ 2 into Eq.#2, and solve for b

Eq. #2 = 3a + 4b = -13

3 * [(b + 17) ÷ 2] + 4b = -13 Multiply 3 * (b + 17)

(3b + 51)÷2 + 4b = -13 Multiply both sides by 2 to eliminate the (÷2)

(3b + 51) + 8b = -26 Add (3b + 8b)

11b + 51 = -26 Add -51 to both sides

11b = -77 Divide both sides by 11

b = -7

Substitute b = -7 into Eq.#1

Eq. #1 = 2a - b = 17

2a + 7 = 17 Subtract 7 from both sides

2a = 10 Divide both sides by 2

a = 5

That Substitution sure was complicated compared to the Elimination method. I wonder if it would have been easier to solve Eq.#1 for a in terms of b. That [(b + 17) ÷ 2] was the trouble maker.

Eq. #1 = 2a - b = 17 Add b to both sides

2a = 17 + b subtract 17 from both sides

2a -17 = b reverse sides

b = 2a - 17

Substitute (b = 2a - 17) into Eq.#2, and solve for a

Eq. #2 = 3a + 4b = -13

3a + 4 * (2a - 17) = -13 Multiply 4 * (2a - 17)

3a + (8a - 68) = -13 Add 3a + 8a

11a - 68 = -13 Add +68 to both sides

11a = 55 Divide by 11

a = 5

Eq. #1 = 2a - b = 17 Substitute a = 5 into Eq.#1

2 * 5 - b = 17 Multiply 2 * 5 = 10

10 - b = 17 Add -10 to both sides

b = -7

We got the same answers as we did by using the elimination method, but the substitution was more work. Sometimes using the elimination method, you have to multiply both equations by 2 different numbers to eliminate a variable, but I still think it is easier.