Special equations

Algebra, grade 7 Lesson 18 of 22Learning

Special equations

Until now every equation had exactly one solution. But in some equations, while solving, $\displaystyle x$ disappears from both sides – and what is left is a statement without $\displaystyle x$ . That statement tells us what the solutions are.

An equation with no solution

\begin{aligned}2(x+3)&=2x+1\\ 2x+6&=2x+1 &&/-2x\\ 6&=1\end{aligned}

The statement $\displaystyle {6}={1}$ is false. Whatever number we substitute, the first side is 5 more than the second – so the equation has no solution.

An equation every number solves

\displaystyle x

is gone: a false statement – no solution; a true statement – every number is a solution.

\begin{aligned}3(x-1)+3&=3x\\ 3x-3+3&=3x\\ 3x&=3x &&/-3x\\ 0&=0\end{aligned}

The statement $\displaystyle {0}={0}$ is always true, so every number is a solution of the equation. No wonder: the two sides are equal expressions, and $\displaystyle {3(x-1)+3}={3x}$ is an identity – as we saw in the lesson about equal expressions.

Checking by substitution

Substituting numbers shows the same. In the equation $\displaystyle {2(x+3)}={2x+1}$ , substituting $\displaystyle {x}={0}$ gives 6 on one side and 1 on the other, and substituting $\displaystyle {x}={10}$ gives 26 and 21 – always a difference of 5. And in the equation $\displaystyle {3(x-1)+3}={3x}$ every substitution gives two equal sides.

Exercise 1

Simplify the side with the brackets, then solve.
An equation has one solution, no solution, or every number is a solution of it.

Exercise 2

Fill in a number for the square so that the equation $\displaystyle {4(x-2)}={4x+\square}$ has no solution.
Now a different number, so that every number is a solution of it.
Can you fill it in so that it has exactly one solution? Explain.

Every

\displaystyle x

/ – 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

Exercise 3

Write an equation of your own that has no solution, and an equation that every number is a solution of.
Swap with a friend and check each other’s equations.

Formulas: algebra, grade 7

Open as a page

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
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$