S2 Mathematics Practice Book · Unit 3: Simultaneous linear equations

Unit 3 Simultaneous linear equations

Key unit competence: Solve problems related to simultaneous linear equations.

Key ideas

A system of two linear equations in $x$ and $y$ is solved by finding the pair $(x, y)$ that satisfies both equations at the same time.
Types of systems: one solution (the lines intersect), no solution (parallel lines), infinitely many solutions (the same line).
Substitution: make one variable the subject of one equation and substitute it into the other.
Elimination: multiply the equations so that one variable has equal (or opposite) coefficients, then subtract (or add).
Comparison: make the same variable the subject of both equations and set them equal.
Cramer's rule for $a x + b y = c$ , $d x + ey = f$ : $Δ = a e - b d$ , $x = \frac{ce - b f}{Δ}$ , $y = \frac{a f - c d}{Δ}$ (if $Δ \neq = 0$ ).
Graphical method: draw both lines; the solution is the point of intersection.
Word problems: name the two unknowns, write two equations from the information, solve, then check in the story.

Formulas to know

Δ = a d b e = a e - b d

x = \frac{c f b e}{Δ}, y = \frac{a d c f}{Δ}

Common misconceptions

"If I find $x$ the system is solved." No: you must also find $y$ and check both in the original equations.
"Parallel lines give infinitely many solutions." No: distinct parallel lines never meet, so there is no solution.

Worked examples

Example 1. Solve $x + y = 10$ and $x - y = 4$ .

Solution

Adding: $2 x = 14$ , $x = 7$ . Then $y = 10 - 7 = 3$ . Solution $(7, 3)$ .

Example 2. 3 pens and 2 books cost 2 100 FRW; 1 pen and 4 books cost 3 200 FRW. Find each price.

Solution

$3 p + 2 b = 2100$ and $p + 4 b = 3200$ . From the second, $p = 3200 - 4 b$ .

$3 (3200 - 4 b) + 2 b = 2100 \Rightarrow 9600 - 10 b = 2100 \Rightarrow b = 750$ , $p = 200$ . A pen costs 200 FRW and a book 750 FRW.

Practice questions

Easier and harder questions are mixed, just like in a real exam.

Solve by substitution: $y = 3 x - 1$ and $2 x + y = 9$ .
Show answer
$2 x + 3 x - 1 = 9$ , $x = 2$ , $y = 5$ .
Common mistake: Forgetting to find $y$ .
The ages of a mother and her daughter add to 50. Five years ago the mother was 4 times as old as her daughter. Find their ages now.
Show answer
$m + d = 50$ and $m - 5 = 4 (d - 5) \Rightarrow m = 4 d - 15$ . $5 d - 15 = 50$ , $d = 13$ , $m = 37$ .
Common mistake: Subtracting 5 from only one age.
Solve by elimination: $5 x + 2 y = 16$ and $3 x - 4 y = - 6$ .
Show answer
Double the first: $10 x + 4 y = 32$ ; add the second: $13 x = 26$ , $x = 2$ ; then $10 + 2 y = 16$ , $y = 3$ . Solution $(2, 3)$ .
Common mistake: Adding equations without first matching the coefficients.
Show that the system $x - 2 y = 3$ and $- 2 x + 4 y = 1$ has no solution.
Show answer
Multiply the first by $- 2$ : $- 2 x + 4 y = - 6$ , but the second says $- 2 x + 4 y = 1$ . Contradiction: parallel lines, no solution.
Common mistake: Concluding there are infinitely many solutions.
Solve by Cramer's rule: $3 x - y = 7$ and $x + 2 y = 0$ .
Show answer
$Δ = 6 + 1 = 7$ ; $x = \frac{14 - 0}{7} = 2$ ; $y = \frac{0 - 7}{7} = - 1$ . Solution $(2, - 1)$ .
Common mistake: Getting the sign of $Δ$ wrong.
Mukamana bought 4 notebooks and 3 pens for 2 600 FRW. Gatete bought 2 notebooks and 5 pens for 2 000 FRW. Find each price.
Show answer
$4 n + 3 p = 2600$ , $2 n + 5 p = 2000$ . Double the second and subtract: $7 p = 1400$ , $p = 200$ , $n = 500$ .
Common mistake: Mixing up which coefficient belongs to which item.
Solve graphically $y = x + 1$ and $y = - x + 5$ .
Show answer
The lines meet at $(2, 3)$ : check $3 = 2 + 1$ and $3 = - 2 + 5$ .
Common mistake: Plotting only one point per line, giving inaccurate lines.
The perimeter of a rectangle is 34 cm and its length is 5 cm more than its width. Find its dimensions.
Show answer
$2 l + 2 w = 34$ , $l = w + 5$ : $4 w + 10 = 34$ , $w = 6$ cm, $l = 11$ cm.
Common mistake: Using $l + w = 34$ .

Matching and drag-and-drop

Matching
Match each item in column A with the correct answer in column B. Write the letter next to each number.
Column A Column B
1. Solve $y = x + 1$ and $y = 3 x - 5$ by comparison. a. $(4, 2)$
2. Solve $2 x + 3 y = 8$ and $x + 4 y = 9$ . b. $(3, 4)$
3. Solve $3 x + 2 y = 7$ and $2 x - 3 y = - 4$ . c. $(1, 2)$
4. Solve $\frac{x}{2} + y = 4$ and $x - y = 2$ . d. $x = 1, y = 2$
Show answer
1 → b ( $(3, 4)$ ) · 2 → c ( $(1, 2)$ ) · 3 → d ( $x = 1, y = 2$ ) · 4 → a ( $(4, 2)$ )
- Step 1. $x + 1 = 3 x - 5$ .
  Step 2. $x = 3$ .
  Step 3. $y = 4$ .
- Step 1. $x = \frac{8 \cdot 4 - 3 \cdot 9}{5} = 1$ .
  Step 2. $y = \frac{2 \cdot 9 - 8 \cdot 1}{5} = 2$ .
- Step 1. Multiply by 3 and 2: $9 x + 6 y = 21$ and $4 x - 6 y = - 8$ . Add: $13 x = 13$ , $x = 1$ .
  Step 2. $3 + 2 y = 7$ , so $y = 2$ .
- Step 1. From the second $x = y + 2$ .
  Step 2. $\frac{y + 2}{2} + y = 4$ gives $y = 2$ .
  Step 3. $x = 4$ .
Drag and drop
Choose the correct word or phrase from the box for each gap. One word in the box is not needed.
$(1, 4.5)$ · has no unique solution · infinitely many solutions · $(2, 3)$
1. The system $x + y = 3$ and $2 x + 2 y = 6$ has .
2. If $Δ = 0$ in Cramer's rule, the system .
3. Solve $3 x + 2 y = 12$ and $5 x - 2 y = 4$ . →
Show answer
1: infinitely many solutions · 2: has no unique solution · 3: $(2, 3)$
- The second equation is twice the first: they are the same line.
- The lines are parallel or the same; there is no unique solution.
- Step 1. Adding: $8 x = 16$ .
  Step 2. $x = 2$ .
  Step 3. Then $6 + 2 y = 12$ .
  Step 4. $y = 3$ .

Column A	Column B
1. Solve $y = x + 1$ and $y = 3 x - 5$ by comparison.	a. $(4, 2)$
2. Solve $2 x + 3 y = 8$ and $x + 4 y = 9$ .	b. $(3, 4)$
3. Solve $3 x + 2 y = 7$ and $2 x - 3 y = - 4$ .	c. $(1, 2)$
4. Solve $\frac{x}{2} + y = 4$ and $x - y = 2$ .	d. $x = 1, y = 2$

Matching

Match each item in column A with the correct answer in column B. Write the letter next to each number.

Column A	Column B
1. Solve $2 x + y = 8$ and $x + y = 5$ .	a. 13 and 7
2. The sum of two numbers is 20 and their difference is 6. What are the numbers?	b. 700 FRW
3. Solve $x + y = 7$ and $x - y = 1$ .	c. $(4, 3)$
4. 3 pens and 2 books cost 2 600 FRW; 1 pen and 3 books cost 2 500 FRW. Find the price of a book.	d. $(3, 2)$

Show answer

1 → d ( $(3, 2)$ ) · 2 → a (13 and 7) · 3 → c ( $(4, 3)$ ) · 4 → b (700 FRW)

Subtract: $x = 3$ ; then $y = 2$ .
Step 1. $x + y = 20$ .
Step 2. $x - y = 6$ gives $x = 13$ .
Step 3. $y = 7$ .
Step 1. Adding gives $2 x = 8$ .
Step 2. So $x = 4$ and $y = 3$ .
Step 1. $3 p + 2 b = 2600$ and $p + 3 b = 2500$ , so $p = 2500 - 3 b$ .
Step 2. $3 (2500 - 3 b) + 2 b = 2600 \Rightarrow 7500 - 7 b = 2600 \Rightarrow b = 700$ FRW (and $p = 400$ ).

Drag and drop
Choose the correct word or phrase from the box for each gap. One word in the box is not needed.
$(3, 6)$ · 2 km/h · $x = 6, y = 3$ · 1 km/h
1. A boat goes 24 km downstream in 2 h and returns upstream in 3 h. Find the speed of the current. →
2. Solve $y = 2 x$ and $x + y = 9$ . →
3. Solve $\frac{x}{2} + \frac{y}{3} = 4$ and $x - y = 3$ . →
Show answer
1: 2 km/h · 2: $(3, 6)$ · 3: $x = 6, y = 3$
- Step 1. $b + c = 12$ and $b - c = 8$ .
  Step 2. So $c = 2$ km/h (boat 10 km/h).
- Step 1. Substitute: $x + 2 x = 9$ .
  Step 2. $x = 3$ .
  Step 3. $y = 6$ .
- Step 1. Multiply the first by 6: $3 x + 2 y = 24$ . From the second, $x = y + 3$ .
  Step 2. $3 (y + 3) + 2 y = 24 \Rightarrow 5 y = 15 \Rightarrow y = 3$ , $x = 6$ .

Take the Unit 3 quiz → · Back to contents