LemonmathSelf-study sheet

Grade 9 · Lesson 1.1

Linear equations and simultaneous linear equations in two unknowns

Name: Date: Time spent: min

Goals

Recognise a linear equation in two unknowns $ax+by=c$ and check whether a pair of numbers is a solution.
Show the solution set of a linear equation in two unknowns as a straight line in the coordinate plane.
Recognise a pair of simultaneous linear equations and check a solution of them.

Remember

Linear equation in two unknowns: $ax+by=c$ with $a\neq0$ or $b\neq0$ .
$(x_0, y_0)$ is a solution when $ax_0+by_0=c$ ; the equation has infinitely many solutions.
The solution set is a straight line: $y=-\dfrac{a}{b}x+\dfrac{c}{b}$ if $b\neq0$ ; $x=\dfrac{c}{a}$ if $b=0$ .
A solution of two simultaneous equations is a pair that satisfies both equations, that is, the point where the two lines meet.

Exercises

Exercise 1

Which of these are linear equations in two unknowns? a) $3x-y=2$ ; b) $0x+5y=-10$ ; c) $x+y^2=4$ ; d) $0x+0y=7$ .

Exercise 2

Which of the pairs $(2, 1)$ , $(0, 4)$ , $(4, -2)$ , $(1, 3)$ are solutions of the equation $3x+2y=8$ ?

Exercise 3

Find $x$ so that $(x, -1)$ is a solution of the equation $4x+3y=9$ .

Exercise 4

Show the solution set of each equation: a) $2x-y=1$ ; b) $0x+2y=6$ .

Exercise 5

A group of $7$ people go to the cinema. A student ticket costs $45$ thousand đồng, an adult ticket costs $70$ thousand đồng, and the group pays $365$ thousand đồng. Let $x$ be the number of students and $y$ the number of adults. Form the simultaneous equations, then say which of $(5, 2)$ and $(4, 3)$ is a solution.

Exercise 6 Review

Write the equation $3x+y=5$ in the form $y=ax+b$ . What is the gradient of the line of solutions, and where does it cross the $y$ -axis?

Exercise 7

Consider the equation $x-2y=4$ .

a) Find the solution with $x=0$ and the solution with $y=0$ .

b) Draw the line showing the solution set of the equation.

c) Of the two solutions $(6, 1)$ and $(2, -1)$ of the equation, which one is also a solution of $2x+y=3$ ?

Exercise 8 Challenge

A shop sells stickers at $3$ thousand đồng a sheet and at $5$ thousand đồng a sheet. Hùng wants at least one sheet of each kind and wants to spend exactly $40$ thousand đồng. Find all the ways he can do it.

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. Which of these is a linear equation in two unknowns?

x^2-y=1

.B.

0x+0y=2

.C.

2x+0y=6

.D.

xy=3

Question 2. Which pair is a solution of the equation

x-3y=2

(5, 1)

.B.

(1, 5)

.C.

(2, 1)

.D.

(-1, 1)

Question 3. The line showing the solution set of the equation

0x+2y=-4

A. the line

y=-2

.B. the line

x=-2

.C. the line

y=2x-4

.D. the line

y=-4

Question 4. Which pair is a solution of the simultaneous equations

\begin{cases} x+y=5\\ 2x-y=1 \end{cases}

(2, 3)

.B.

(3, 2)

.C.

(1, 4)

.D.

(1, 1)

Part II. True or false

Question 1. Consider the equation

2x+y=7

a) $(3, 1)$ is a solution of the equation.	T / F
b) The equation has exactly one solution.	T / F
c) The line showing the solution set passes through the point $(0, 7)$ .	T / F
d) $(3, 1)$ is also a solution of the simultaneous equations $\begin{cases} 2x+y=7\\ x-y=1 \end{cases}$ .	T / F

Part III. Short answer

Question 1. Consider the equation

3x-2y=12

. Find

y

so that

(2, y)

is a solution.

Question 2. How many solutions

(x, y)

of the equation

x+2y=9

have both

x

and

y

positive whole numbers?

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $ax+by=c$ is a linear equation in two unknowns when $a\neq0$ or $b\neq0$ , and $x$ , $y$ appear with no power other than $1$ .

Steps:

Read off the coefficients of $x$ and $y$ in each equation.
Rule out the equation with $y^2$ and the one where both coefficients are $0$ .

Exercise 2.

What you need: $(x_0, y_0)$ is a solution when $3x_0+2y_0=8$ .

Steps:

Substitute each pair into $3x+2y$ .
Compare the result with $8$ .

Exercise 3.

What you need: substituting the known value turns the equation into a linear equation in one unknown.

Steps:

Substitute $y=-1$ .
Solve the equation in one unknown that you get.

Exercise 4.

What you need: when $b\neq0$ , write $y$ in terms of $x$ and draw the line through two points.

Steps:

Rewrite in the form $y=\dots$ .
Find two points on the line (for example with $x=0$ and $x=1$ ).

Exercise 5.

What you need: each condition in the problem gives one equation; a solution satisfies both.

Steps:

The number of people gives $x+y=7$ .
The cost gives $45x+70y=365$ .
Substitute each pair into both equations.

Exercise 6.

What you need: the line $y=ax+b$ has gradient $a$ and crosses the $y$ -axis at $(0, b)$ (Grade 8).

Steps:

Move $3x$ to the right-hand side.
Read off $a$ and $b$ .

Exercise 7.

What you need: two solutions give two points of the line; a common solution of two equations solves the simultaneous equations.

Steps:

Put $x=0$ , then $y=0$ .
Draw the line through the two points you found.
Substitute each pair into $2x+y$ .

Exercise 8.

What you need: a way of buying is a solution $(x, y)$ of $3x+5y=40$ with $x$ , $y$ positive whole numbers.

Steps:

Write $5y=40-3x$ ; since $40$ and $5y$ are multiples of $5$ , so is $3x$ .
So $x$ is a multiple of $5$ ; try $x=5, 10, 15$ .

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