An equation and its solution

Algebra, grade 7 Lesson 16 of 22Learning

An equation and its solution

An equation is two expressions with an equals sign between them, for example $\displaystyle {2x+3}={13}$ . The equation asks a question: for which number $\displaystyle x$ are the two expressions equal?

Sides

Every equation has two sides: the expression to the left of the equals sign and the one to its right. In the equation $\displaystyle {5x-1}={3x+7}$ one side is $\displaystyle 5x-1$ and the other is $\displaystyle 3x+7$ . Here the variable $\displaystyle x$ is called the unknown – the number we are looking for.

A solution

A solution – a number that makes the equation a true statement.

A number is a solution of the equation if, when we substitute it for $\displaystyle x$ , the two sides come out equal. In the equation $\displaystyle {2x+3}={13}$ : substituting $\displaystyle {x}={5}$ gives $\displaystyle {2\cdot5+3}={13}$ – a true statement, so 5 is a solution. Substituting $\displaystyle {x}={4}$ gives $\displaystyle {2\cdot4+3}={11}$ , and 11 is not 13 – so 4 is not a solution.

Checking by substitution

To check whether a number is a solution, substitute it into each side separately, calculate both sides and compare. When $\displaystyle x$ appears on both sides, substitute the same number into both.

Exercise 1

Check by substitution whether the given number is a solution of the equation.
Calculate each side, then write your answer.

$\displaystyle {3x-7}={5}$ : is $\displaystyle {x}={4}$ a solution? (The other side is the number 5.)

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

$\displaystyle {2x+8}={x+5}$ : is $\displaystyle {x}={-3}$ a solution?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

$\displaystyle {5-2x}={x+4}$ : is $\displaystyle {x}={-1}$ a solution?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Expression, equation and identity

In an identity every number is a solution.

$\displaystyle 4x-x$ is an expression: it has a value for every $\displaystyle x$ , but there is no question in it.
$\displaystyle {4x-x}={9}$ is an equation: it is true only for $\displaystyle {x}={3}$ .
$\displaystyle {4x-x}={3x}$ is an identity: the two expressions are equal for every number – as we saw in the lesson about equal expressions.

Exercise 2

Which is each of these – an expression, an equation or an identity? Explain.
$\displaystyle \quad$ a. $\displaystyle {7+x}={x+7}$ $\displaystyle \quad$ b. $\displaystyle 2(x-1)$ $\displaystyle \quad$ c. $\displaystyle {2(x-1)}={8}$ $\displaystyle \quad$ d. $\displaystyle {x\cdot x}={x^2}$

Exercise 3

Find a solution of each equation without a long calculation – just by thinking and checking: $\displaystyle {x+7}={12}$ , $\displaystyle {3x}={-12}$ , $\displaystyle {10-x}={13}$ .
The equation $\displaystyle {x^2}={9}$ has two solutions – find both.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: algebra, grade 7

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Adding and subtracting signed numbers

The sum of two opposite numbers
$\displaystyle (-a)+a$ $\displaystyle =$ $\displaystyle 0$
Subtracting a number is adding its opposite
$\displaystyle a-b$ $\displaystyle =$ $\displaystyle a+(-b)$
Subtracting a negative number
$\displaystyle a-(-b)$ $\displaystyle =$ $\displaystyle a+b$

Multiplying and dividing signed numbers

Positive times negative – negative
$\displaystyle a\cdot(-b)$ $\displaystyle =$ $\displaystyle -(a\cdot b)$
Negative times negative – positive
$\displaystyle (-a)\cdot(-b)$ $\displaystyle =$ $\displaystyle a\cdot b$
Dividing: different signs – the quotient is negative
$\displaystyle (-a):b$ $\displaystyle =$ $\displaystyle -(a:b)$ $\displaystyle b\neq0$
Dividing: the same sign – the quotient is positive
$\displaystyle (-a):(-b)$ $\displaystyle =$ $\displaystyle a:b$ $\displaystyle b\neq0$

Powers and roots

A power – $\displaystyle a$ the base, $\displaystyle n$ the exponent: how many times $\displaystyle a$ is multiplied
$\displaystyle a^{n}$ $\displaystyle =$ $\displaystyle \underbrace{a\cdot a\cdot\ldots\cdot a}_{n}$
A number to the power 1 is the number itself
$\displaystyle a^{1}$ $\displaystyle =$ $\displaystyle a$
A negative base with an even exponent – the power is positive
$\displaystyle (-a)^{2}$ $\displaystyle =$ $\displaystyle a^{2}$
The square root – the positive number (or 0) whose square is the number under the root
$\displaystyle \sqrt{a^{2}}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$

The laws of arithmetic and like terms

The commutative law of addition
$\displaystyle a+b$ $\displaystyle =$ $\displaystyle b+a$
The commutative law of multiplication
$\displaystyle a\cdot b$ $\displaystyle =$ $\displaystyle b\cdot a$
The associative law of addition
$\displaystyle (a+b)+c$ $\displaystyle =$ $\displaystyle a+(b+c)$
The associative law of multiplication
$\displaystyle (a\cdot b)\cdot c$ $\displaystyle =$ $\displaystyle a\cdot(b\cdot c)$
A number times itself – squared
$\displaystyle x\cdot x$ $\displaystyle =$ $\displaystyle x^{2}$
Collecting like terms – add the coefficients; the letters stay
$\displaystyle ax+bx$ $\displaystyle =$ $\displaystyle (a+b)x$

The distributive law

The distributive law
$\displaystyle a(b+c)$ $\displaystyle =$ $\displaystyle ab+ac$
The distributive law over a difference
$\displaystyle a(b-c)$ $\displaystyle =$ $\displaystyle ab-ac$
A minus before brackets – every term inside changes its sign
$\displaystyle -(a-b)$ $\displaystyle =$ $\displaystyle -a+b$

Distance, speed and time

$\displaystyle s$ the distance, $\displaystyle v$ the speed, $\displaystyle t$ the time
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$