Linear equations in N, Z, D and Q

Does the equation $2x+9=1$ have a solution? Yes if we look in $\mathbb{Z}$: $x=-4$. No if we look only among natural numbers. So the answer to an equation depends on the set in which we solve it. In this lesson you solve first-degree equations in $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{D}$ and $\mathbb{Q}$, as well as equations that reduce to them, and you turn word problems into equations.

Solving in a given set

First solve the equation, then keep only the solutions that belong to the required set. We write $S$ for the solution set.

The symbol $\varnothing$ is the empty set: no solution.

Special cases after simplifying

After expanding and grouping, we reach an equation of the form $ax=b$.

• If $a\ne0$: exactly one solution, $x=\dfrac{b}{a}$.
• If $a=0$ and $b\ne0$: the equality $0=b$ is false, the equation is impossible: $S=\varnothing$.
• If $a=0$ and $b=0$: the equality $0=0$ is always true, every number in the set is a solution.

Equations that reduce to first degree:

• when you expand, the terms in $x^2$ cancel: $(x+1)^2=x^2+5$ becomes $2x+1=5$;
• a product equation $(ax+b)(cx+d)=0$ gives two first-degree equations.

Turning a problem into an equation

1. Choose the unknown and say what it stands for, with its unit.
2. Translate each sentence of the problem using the unknown.
3. Write the equation and solve it.
4. Check that the solution makes sense: a number of people must be a natural number, a length must be positive.
5. Answer with a sentence.