Sec 2 · G1 (N(T))

Linear equations, inequalities and simultaneous equations

Practise linear equations, inequalities and simultaneous equations with step-by-step help. Free account.

1. Graphs of linear equations

Equation	Gradient	$y$ -intercept
$y = mx + c$	$m$	$c$
$2x + 3y = 12$ , so $y = -\dfrac{2}{3}x + 4$	$-\dfrac{2}{3}$	$4$
$x = 3$	undefined (vertical line)	none
$y = -2$	$0$ (horizontal line)	$-2$

Where 2x + 3y = 12 cuts the axes

\displaystyle 2x + 3y = 12

$x = 0$ : $3y = 12$ , so $(0, 4)$
$y = 0$ : $2x = 12$ , so $(6, 0)$

📖 Worked example

Find the $y$ -intercept of the line $-2x - y = -8$ .

Step 1 of 3

Make y the subject: move the x term across (it changes sign)

-y = 2x - 8

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2. Simultaneous equations

The lines y = 2x − 1 and x + y = 5 cross at (2, 3), the solution of both equations.

Substitution: y = 2x − 1 into x + y = 5

\displaystyle x + (2x - 1) = 5

e.g.

\displaystyle x + (2x - 1) = 5

\displaystyle 3x - 1 = 5

\displaystyle x = 2

$y = 2(2) - 1$ , which is $3$

Elimination: add to remove y

\displaystyle 3x + 2y = 16 \text{ and } 5x - 2y = 16

e.g.

\displaystyle 8x = 32

\displaystyle x = 4

$3(4) + 2y = 16$ , so $y = 2$

📖 Worked example

Solve the simultaneous equations $-6x - 3y = 24$ and $x + 3y = -14$ .

Step 1 of 4

Number the equations

-6x - 3y = 24

... (1)

x + 3y = -14

... (2)

📖 Worked example

Solve the simultaneous equations $-3x - 3y = -33$ and $y = 3x - 9$ by substitution.

Step 1 of 4

Number the equations

-3x - 3y = -33

... (1)

y = 3x - 9

... (2)

📖 Worked example

3 curry puffs and 1 packet of soya bean milk cost $4.70. 1 curry puff and 2 packets of soya bean milk cost $3.40. Find the cost of one curry puff and the cost of one packet of soya bean milk.

Step 1 of 6

Let one curry puff cost x dollars and one packet of soya bean milk cost y dollars. The first purchase

3x + y = 4.7

... (1)

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1. Graphs of linear equations

Equation	Gradient	$y$ -intercept
$y = mx + c$	$m$	$c$
$2x + 3y = 12$ , so $y = -\dfrac{2}{3}x + 4$	$-\dfrac{2}{3}$	$4$
$x = 3$	undefined (vertical line)	none
$y = -2$	$0$ (horizontal line)	$-2$

Where 2x + 3y = 12 cuts the axes

\displaystyle 2x + 3y = 12

$x = 0$ : $3y = 12$ , so $(0, 4)$
$y = 0$ : $2x = 12$ , so $(6, 0)$

📖 Worked example

Find the $y$ -intercept of the line $-2x - y = -8$ .

Step 1 of 3

Make y the subject: move the x term across (it changes sign)

-y = 2x - 8

Practise this →

2. Simultaneous equations

Substitution: y = 2x − 1 into x + y = 5

\displaystyle x + (2x - 1) = 5

e.g.

\displaystyle x + (2x - 1) = 5

\displaystyle 3x - 1 = 5

\displaystyle x = 2

$y = 2(2) - 1$ , which is $3$

Elimination: add to remove y

\displaystyle 3x + 2y = 16 \text{ and } 5x - 2y = 16

e.g.

\displaystyle 8x = 32

\displaystyle x = 4

$3(4) + 2y = 16$ , so $y = 2$

📖 Worked example

Solve the simultaneous equations $-6x - 3y = 24$ and $x + 3y = -14$ .

Step 1 of 4

Number the equations

-6x - 3y = 24

... (1)

x + 3y = -14

... (2)

📖 Worked example

Solve the simultaneous equations $-3x - 3y = -33$ and $y = 3x - 9$ by substitution.

Step 1 of 4

Number the equations

-3x - 3y = -33

... (1)

y = 3x - 9

... (2)

📖 Worked example

3 curry puffs and 1 packet of soya bean milk cost $4.70. 1 curry puff and 2 packets of soya bean milk cost $3.40. Find the cost of one curry puff and the cost of one packet of soya bean milk.

Step 1 of 6

Let one curry puff cost x dollars and one packet of soya bean milk cost y dollars. The first purchase

3x + y = 4.7

... (1)

Practise this →