Linear equations for CDS

Linear equations are the plainest part of CDS algebra, and that is exactly why they matter. Many questions that look like arithmetic (ages, digits, coins, fractions) are really a pair of linear equations. The paper also asks directly about the number of solutions a pair of equations has, which is a pure ratio check once you know the table.

The skills to build are simple: translate words into equations without error, solve by the quickest route, and verify by substitution. Verification takes five seconds and catches most sign slips.

One variable

Move all terms in x to one side and the numbers to the other, then divide. Whatever you do to one side, do to the other.

• 3x − 7 = 11 gives 3x = 18 and x = 6.
• x/2 + x/3 = 10 gives 5x/6 = 10 and x = 12.

Two variables: three methods

Elimination

Multiply the equations so that one variable has equal and opposite coefficients, then add.

• 3x + 4y = 10 and 2x − 3y = 1
• Multiply the first by 3 and the second by 4: 9x + 12y = 30 and 8x − 12y = 4
• Adding: 17x = 34, so x = 2, and then y = 1

Substitution

When one equation gives a variable directly, such as y = 2x − 5, put it into the other equation. This is fastest when a coefficient is already 1.

Swapped coefficients

When the coefficients are swapped, as in ax + by = p and bx + ay = q, add and subtract the equations.

• Adding: (a + b)(x + y) = p + q
• Subtracting: (a − b)(x − y) = p − q

This gives x + y and x − y at once, and the rest is one step. It turns ugly-looking pairs like 37x + 43y = 123 and 43x + 37y = 117 into easy ones.

Equations in 1/x and 1/y

If x and y appear only as 1/x and 1/y, put u = 1/x and v = 1/y. Solve the linear pair in u and v, then take reciprocals.

How many solutions?

For a₁x + b₁y = c₁ and a₂x + b₂y = c₂, compare the ratios of the coefficients.

Why: equal a and b ratios mean the lines have the same slope. Then they are either the same line (c ratio also equal) or never meet.

Translating word problems

Worked questions

Question 1: The sum of the digits of a two-digit number is 9. Reversing the digits increases the number by 27. Find the number.

• Let the number be 10a + b. Then a + b = 9.
• (10b + a) − (10a + b) = 27, so 9(b − a) = 27 and b − a = 3.
• b = 6 and a = 3, so the number is 36. Check: 63 − 36 = 27.

Question 2: Five years ago, a parent was 7 times as old as their child. Five years from now, the parent will be 3 times as old as the child. Find their present ages.

• Let the ages be P and C. Then P − 5 = 7(C − 5), so P = 7C − 30.
• P + 5 = 3(C + 5), so 7C − 30 + 5 = 3C + 15, which gives 4C = 40.
• Child = 10 years, parent = 40 years. Check: 5 years ago, 35 = 7 × 5. In 5 years, 45 = 3 × 15.

Question 3: If 1 is added to both the numerator and the denominator of a fraction, it becomes 4/5. If 5 is subtracted from both, it becomes 1/2. Find the fraction.

• (x + 1)/(y + 1) = 4/5 gives 5x − 4y = −1.
• (x − 5)/(y − 5) = 1/2 gives y = 2x − 5.
• Substituting: 5x − 8x + 20 = −1, so x = 7 and y = 9. The fraction is 7/9.

Question 4: Solve 2/x + 3/y = 13 and 5/x − 4/y = −2.

• Put u = 1/x and v = 1/y: 2u + 3v = 13 and 5u − 4v = −2.
• Multiply by 4 and 3: 8u + 12v = 52 and 15u − 12v = −6. Adding gives 23u = 46, so u = 2 and v = 3.
• So x = 1/2 and y = 1/3.

Question 5: Solve 3x + 5y = 21 and 5x + 3y = 27.

• Adding: 8(x + y) = 48, so x + y = 6.
• Subtracting the first from the second: 2(x − y) = 6, so x − y = 3.
• x = 4.5 and y = 1.5. Check: 13.5 + 7.5 = 21 and 22.5 + 4.5 = 27.

Question 6: For what value of k do 2x + 3y = 7 and (k − 1)x + (k + 2)y = 3k have infinitely many solutions?

• (k − 1)/2 = (k + 2)/3 gives 3k − 3 = 2k + 4, so k = 7.
• Check the constant ratio: 3k/7 = 21/7 = 3, and the other ratios are 6/2 = 3 and 9/3 = 3.
• So k = 7.

Practice set

1. Solve 3x − 7 = 11.
2. Solve x + y = 12 and 2x − y = 3.
3. For what value of k do kx + 2y = 3 and 6x + 4y = 5 have no solution?
4. A purse has only ₹5 and ₹10 notes, 25 notes in all, worth ₹175. How many of each are there?
5. A two-digit number is 4 times the sum of its digits. If 18 is added to it, the digits are reversed. Find the number.
6. Solve 37x + 43y = 123 and 43x + 37y = 117.
7. For what value of k do kx + 3y = k − 3 and 12x + ky = k have infinitely many solutions?
8. The ages of a parent and a child add up to 50. Five years ago, the parent was 4 times as old as the child. Find their present ages.

Answers

1. x = 6. 3x = 18.
2. x = 5, y = 7. Adding gives 3x = 15.
3. k = 3. k/6 = 2/4, and then 2/4 ≠ 3/5, so there is no solution.
4. 15 notes of ₹5 and 10 notes of ₹10. x + y = 25 and 5x + 10y = 175, so x + 2y = 35 and y = 10.
5. 24. 10a + b = 4(a + b) gives b = 2a. Reversal gives b − a = 2. So a = 2, b = 4.
6. x = 1, y = 2. Adding gives x + y = 3. Subtracting gives x − y = −1.
7. k = 6. k/12 = 3/k gives k = ±6. For k = 6 all ratios are 1/2. For k = −6 the constant ratio is 3/2, so there is no solution.
8. Parent 37, child 13. P = 4C − 15 and P + C = 50, so 5C = 65. Check: 32 = 4 × 8.

What to do next

• Check every answer by substitution for the next two weeks, until it becomes a habit
• Practise five swapped-coefficient pairs and five 1/x, 1/y pairs
• Learn the consistency table and test both values of k every time
• Move on to quadratic equations, and revise ratio and proportion for the word problems