Equations Reducible to a Pair of linear equations in two variables

Example 12

Solve the following pair of equations.

2x + 3y = 13

5x - 4y = -2

Solution: Observe the given pair of equations. They are not linear equations. (Why?)

We have 2(1x) + 3(1y) = (1)

5(1x) - 4(1y) = (2)

If we substitute 1x = p and 1y = q, we get the following pair of linear equations:

2p + 3q = 13 (3)

5p - 4q = -2 (4)

Coefficients of q are 3 and 4 and their LCM is 12. Using the elimination method:

Equation (3) × 4 8p + 12q =

Equation (4) × 3 15p - 12q = 'q' terms have opposite sign, so we add the two equations

23p =

Substitute the value of p in equation (3)

2(2) + 3q = 13

4 + 3q =

3q =

q =

Since p = 1x and q = 1y

1x = and 1y =

Therefore:

x = 1213

y = 1312

Example 13

Kavitha thought of constructing 2 more rooms in her house. She enquired about the labour. She came to know that 6 men and 8 women could finish this work in 14 days. But she wishes to complete that work in only 10 days. When she enquired, she was told that 8 men and 12 women could finish the work in 10 days. Find out how much time would be taken to finish the work if one man or women worked alone.

Solution:

6 men and 8 women can complete the work in days.

8 men and 12 women can complete the work in days.

We need to find out how long it would take one man alone and one woman alone to complete the same work.

Let's use variables:

Let 'x' be the number of days one man takes to complete the work alone.

Let 'y' be the number of days one woman takes to complete the work alone.

Formulating Equations

Work done by one man in one day = 1x

Work done by one woman in one day = 1y

Using the information from the scenarios, we can set up two equations:

Scenario 1: 6 men and 8 women can complete the work in 14 days.

Work done by 6 men in one day = 6x1x

Work done by 8 women in one day = 8y1y

Combined work done in one day = (6x) + (8y)

Since they complete the work in 14 days, the equation is: (6x) + (8y) = 114

8 men and 12 women can complete the work in 10 days.

Work done by 8 men in one day = 8x

Work done by 12 women in one day = 12y

Combined work done in one day = (8x) + (12y)

Since they complete the work in 10 days, the equation is: (8x) + (12y) = 110

Solving the Equations

We now have a system of two equations with two variables:

(6x) + (8y) = 114

(8x) + (12y) = 110

To make it easier, let's substitute:

Let a = 1x

Let b = 1y

Our equations become:

6a + 8b = 114

8a + 12b = 110

Now, we can solve for 'a' and 'b' using any method (substitution or elimination). Let's use elimination:

Multiply equation (1) by 4:

24a + 32b = 414 Multiply equation (2) by 3:

24a + 36b = 310210

Subtract the second equation from the first:

-4b = (414) - (310)

-4b = (2070) - (2170)

-4b = −170−150

b = 1280 1230

Now substitute the value of 'b' back into equation (1):

6a + 8(1280) = 114

6a + 135 = 114

6a = (114) - (135)

6a = (570) - (270)

6a = 370270

a = 11403140

Finding x and y

Remember that:

a = 1x => x = 1a => x = 1140

b = 1y => y = 1b => y = 1280

It would take one man 140 days to finish the work alone.

It would take one woman 280 days to finish the work alone.