2.1 Solving linear equations

Equations and inequalities: solving linear equations

The simplest equation to solve is a linear equation. A linear equation is an equation where the power of the variable(letter, e.g. $x$ ) is 1(one). The following are examples of linear equations.

In this section, we will learn how to find the value of the variable that makes both sides of the linear equation true. For example, what value of $x$ makes both sides of the very simple equation, $x+1=1$ true.

Since the definition of a linear equation is that if the variable has a highest power of one (1), there is at most one solution or root for the equation.

This section relies on all the methods we have already discussed: multiplying out expressions, grouping terms and factorisation. Make sure that you arecomfortable with these methods, before trying out the work in the rest of this chapter.

Now we see that $2x=-1$ . This means if we divide both sides by 2, we will get:

If we substitute $x=-\frac{1}{2}$ , back into the original equation, we get:

That is all that there is to solving linear equations.

Method: solving linear equations

The general steps to solve linear equations are:

1. Expand brackets Expand (Remove) all brackets that are in the equation.
2. Rearrange "Move" all terms with the variable to the left hand side of the equation, and all constant terms (the numbers) to the right hand side of the equals sign.Bearing in mind that the sign of the terms will change from ( $+$ ) to ( $-$ ) or vice versa, as they "cross over" the equals sign.
3. Group like terms Group all like terms together and simplify as much as possible.
4. Factorise If necessary factorise.
5. Write solution Find the solution and write down the answer(s).
6. Check Substitute solution into original equation to check answer.

Solving linear equations

1. Solve for $y$ : $2y-3=7$
2. Solve for $y$ : $-3y=0$
3. Solve for $y$ : $4y=16$
4. Solve for $y$ : $12y+0=144$
5. Solve for $y$ : $7+5y=62$
6. Solve for $x$ : $55=5x+\frac{3}{4}$
7. Solve for $x$ : $5x=3x+45$
8. Solve for $x$ : $23x-12=6+2x$
9. Solve for $x$ : $12-6x+34x=2x-24-64$
10. Solve for $x$ : $6x+3x=4-5(2x-3)$
11. Solve for $p$ : $18-2p=p+9$
12. Solve for $p$ : $\frac{4}{p}=\frac{16}{24}$
13. Solve for $p$ : $\frac{4}{1}=\frac{p}{2}$
14. Solve for $p$ : $-(-16-p)=13p-1$
15. Solve for $p$ : $6p-2+2p=-2+4p+8$
16. Solve for $f$ : $3f-10=10$
17. Solve for $f$ : $3f+16=4f-10$
18. Solve for $f$ : $10f+5+0=-2f+-3f+80$
19. Solve for $f$ : $8(f-4)=5(f-4)$
20. Solve for $f$ : $6=6(f+7)+5f$