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Linear Equations (Cambridge (CIE) IGCSE Maths: Core): Flashcards

Exam code: 0580 & 0980

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FrontSolving Linear Equations
Define a linear equation.

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Cards in this collection (24)

Define a linear equation.
A linear equation is one that can be written in the form $a x + b = c$ , where $a$ , $b$ and $c$ are numbers and $x$ is the variable.
The greatest power of $x$ is $1$ , so a linear equation contains no $x^{2}$ term.
When solving $2 x + 1 = 9$ , why must you subtract $1$ from both sides?
Because the two sides are equal, and they stay equal only if exactly the same thing is done to each of them.
Subtracting $1$ from both sides leaves $2 x = 8$ , which is easier to solve.
Complete the pair of sentences by filling in the missing operations:
Addition is undone by $______$ .
Multiplication is undone by $______$ .
The completed sentences are:
Addition is undone by subtraction.
Multiplication is undone by division.
Two operations that undo each other are called inverse operations.
Solve $5 x - 8 = 22$ .
$x = 6$ .
Adding $8$ to both sides gives $5 x = 30$ , and then dividing both sides by $5$ gives $x = 6$ .
True or False?
Dividing both sides of $4 x + 8 = 12$ by $4$ gives $x + 8 = 3$ .
False.
Every term has to be divided by $4$ , including the $8$ , so the correct line is $x + 2 = 3$ .
It is usually easier to subtract the $8$ first, which leaves $4 x = 4$ .
How do you start solving $2 - 3 x = 10$ , where the $x$ term is negative?
Add $3 x$ to both sides, which removes the negative term and gives $2 = 10 + 3 x$ .
Alternatively, rewrite the left-hand side as $- 3 x + 2$ and carry on from there, since $2 - 3 x$ and $- 3 x + 2$ are the same thing.
To solve $2 (x - 3) = 10$ you can expand the bracket first. What is the alternative, and what is its drawback?
You can divide both sides by $2$ first, giving $x - 3 = 5$ , which works cleanly because the whole left-hand side is multiplied by $2$ .
The drawback is that dividing can produce awkward fractions whenever the number outside the bracket does not divide the other side exactly.
True or False?
Multiplying both sides of $\frac{x}{5} + 4 = \frac{9}{2}$ by $10$ gives $2 x + 4 = 45$ .
False.
The $4$ has to be multiplied by $10$ as well, because every term on both sides must be multiplied.
The correct line is $2 x + 40 = 45$ .
An equation contains fractions with denominators $4$ and $6$ . What should you multiply both sides by?
Multiply both sides by the lowest common denominator, which is $12$ here.
Multiplying by it clears every fraction in one step, because both $4$ and $6$ divide into $12$ exactly.
Complete the working. Both sides of $\frac{5 x}{4} = \frac{1}{2}$ have been multiplied by $4$ :
$______x =______$
The completed line is:
$5 x = 2$
On the left the two $4$ s cancel, and on the right $4 \times \frac{1}{2} = 2$ .
How do you start solving $\frac{4}{x - 2} = 3$ , where the unknown is in the denominator?
Multiply both sides by that denominator, $(x - 2)$ , which gives $4 = 3 (x - 2)$ .
The unknown is now out of the denominator, so the bracket can be expanded and the equation solved as usual.
An equation has $x$ terms on both sides. What must you do before it can be solved in the usual way?
Collect all the $x$ terms onto one side of the equation.
You do this by choosing the side you want to clear and applying the opposite of its $x$ term to both sides.
Complete the working. Both sides of $4 x - 7 = 11 + x$ have had $x$ subtracted:
$______x - 7 =______$
The completed line is:
$3 x - 7 = 11$
The $4 x$ has become $3 x$ , and there is no longer an $x$ term on the right-hand side.
In $4 - 5 x = 6 x - 29$ , which $x$ term should you remove, and why?
Remove the $- 5 x$ , by adding $5 x$ to both sides.
Its coefficient, $- 5$ , is the smaller of the two, so removing it leaves a positive $x$ term, $11 x$ , on the right.
True or False?
Once the $x$ terms have been collected onto one side, the equation is solved in exactly the same way as one with $x$ on only one side.
True.
Collecting the $x$ terms is the only extra step, and after it the equation has the same form as any other linear equation.
It is then solved by undoing the remaining operations one at a time.
You solve an equation and reach $3 = x$ . How should the final answer be written?
Write it as $x = 3$ .
The two sides of an equation can be swapped without changing it, so $3 = x$ is already correct; giving the answer with the letter on the left is simply the usual convention.
Use algebra to represent double the amount of a quantity, given that $x$ is the initial quantity.
Double the amount of a quantity $x$ can be written as $2 x$ .
Use algebra to represent half the amount of a quantity, given that $x$ is the initial quantity.
Half the amount of a quantity $x$ can be written as $\frac{1}{2} x$ or $\frac{x}{2}$ or $0.5 x$ .
Use algebra to represent five more than a quantity, given that $x$ is the initial quantity.
Five more than a quantity $x$ can be written as $x + 5$ .
This is not the same as $5 x$ .
Use algebra to represent ten lots of a quantity, given that $x$ is the initial quantity.
Ten lots of a quantity $x$ can be written as $10 x$ .
This is not the same as $x + 10$ .
True or False?
If my sister is 2 years younger than me, then $x$ is my age and $x - 2$ is her age.
True.
If my sister is 2 years younger than me, then $x$ is my age and $x - 2$ is her age.
It is also true to say that $y + 2$ is my age and $y$ is her age.
Three bags contain $x$ , $x + 5$ and $x + 10$ marbles each.
If there are 75 marbles in total, explain how to form an equation.
Three bags contain $x$ , $x + 5$ and $x + 10$ marbles each.
If there are 75 marbles in total, then an equation can be formed by adding all the quantities up and setting them equal to 75.
This gives $x + (x + 5) + (x + 10) = 75$ .
Examiners want to see this full equation (before simplifying or solving).
If the total perimeter of the regular pentagon below is $100$ cm, show how to form an equation.
The pentagon being regular means the side lengths are all equal.
If the total perimeter of a regular pentagon with side length $(x + 2)$ is $100$ cm, then:
1. Find the total perimeter in terms of $x$ by multiplying $(x + 2)$ by 5
2. Set this equal to 100
This forms the equation $5 (x + 2) = 100$ .
True or False?
If the interior angles of an $n$ -sided polygon are given algebraically in terms of $x$ , you can calculate the size of $x$ as the total sum of the angles will always be $180 (n - 2)$ .
True.
If the interior angles of an $n$ -sided polygon are given algebraically in terms of $x$ , you can calculate the size of $x$ as the total sum of the angles will always be $180 (n - 2)$ .
E.g. for any hexagon (regular or irregular), the sum of its interior angles will always be $180 (6 - 2) = 720 °$ .
So if the interior angles are in terms of $x$ , add them together and set equal to $720 °$ .

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