Section 1.3
Linear Equations 1
If an equation is linear, collect all terms with the variable (typically, $x$) on one side of the equation and all the other terms on the other side. Then divide both sides of the equation by the coefficient of the variable $x$.
As a general rule, collect the variables on whichever side of the equation has more of them. In example (b.) below, on the right is "more $x$'s" than $2x$ on the left. And in example (c.) below, $-5x$ on the right is more $x$'s than $-7x$ on the left.
Examples:
a. \begin{align}
2x-7&=5&&\\
2x&=5+7&&\text{add 7 to both sides}\\
2x&=12&&\text{combine like terms}\\
x&=6&&\text{divide both sides by 2}
\end{align}
b. \begin{align}
2x-6&=6(x+1)&&\\
2x-6&=6x+6&&\text{distribute the 6}\\
-12&=4x&&\text{add $-6$ and $-2x$ to both sides}\\
-3&=x&&\text{divide both sides by 4}
\end{align}
c. \begin{align}
7(2-x)&=4-5x&&\\
14-7x&=4-5x&&\text{distribute the 7}\\
10&=2x&&\text{add $7x$ and $-4$ to both sides}\\
5&=x&&\text{divide both sides by 2}$
\end{align}
d. \begin{align}
x(x+3)&=5+x(x-7)&&\\
x^2+3x&=5+x^2-7x&&\text{distribute $x$}\\
10x&=5&&\text{add $-x^2$ and $7x$ to both sides}\\
x&=\frac{1}{2}&&\text{divide by 2}
\end{align}
When solving for $x$ in terms of $a$, what looks like a variable ($a$) should be thought of as a constant whose value does not happen to be known at the moment. Again, any term that contains the variable $x$ should be isolated on one side of the equation; and all terms that do not have the variable $x$ are collected on the other side.
e. \begin{align}
3x+2(x-a)&=7+a\\
3x+2x-2a&=7+a&&\text{distribute the 2}\\
5x&=7+3a&&\text{simplify like terms and add $2a$ to both sides}\\
x&=\frac{7+3a}{5}&&\text{divide both sides by 5}
\end{align}
f. \begin{align}
5x-2(x+3a)&=7+a(x+1)\\
5x-2x-6a&=7+ax+a&&\text{distribute $-2$ and $a$}\\
3x-ax&=7+7a&&\text{simplify like terms and add $-a$ and $-ax$ to both sides}\\
x(3-a)&=7+7a&&\text{factor out an $x$}\\
x&=\frac{7+7a}{3-a}&&\text{divide both sides by $3-a$}
\end{align}
Finally, if there are fractions in the equation, multiplying both sides of the equation by the least common denominator will eliminate the fractions.
g. \begin{align}
\frac{x}{3}+\frac{2}{5}&=\frac{7x}{15}\\
15\left(\frac{x}{3}+\frac{2}{5}\right)&=\left(\frac{7x}{15}\right)15&&\text{mulitply both sides of the equation by the LCD}\\
15\cdot \frac{x}{3}+15\cdot \frac{2}{5}&=\frac{7x}{15}\cdot15&&\text{distribute 15}\\
5x+6&=7x&&\text{simplify}\\
6&=2x&&\text{add $5x$ to both sides}\\
3&=x&&\text{divide both sides by 2}
\end{align}
h. \begin{align}
\frac{2x}{3}-\frac{x+1}{6}&=\frac{3}{2}\\
6\left(\frac{2x}{3}-\frac{x+1}{6}\right)&=\left(\frac{3}{2}\right)6&&\text{multiply both sides by LCD}\\
6\cdot\frac{2x}{3}-6\cdot\frac{x+1}{6}&=\frac{3}{2}\cdot 6&&\text{distribute 6}\\
4x-(x+1)&=9&&\text{simplify}\\
4x-x-1&=9&&\text{distribute $-1$}\\
3x&=10&&\text{combine like terms}\\
x&=\frac{10}{3}&&\text{divide both sides by 3}
\end{align}
Exercises:
Solve each of the following equations for $x$.
1. $3x+2=8$
2. $5x-4=11$
3. $2(3x-5)=8$
4. $4(x+1)=3$
5. $3(x+4)=5(x-1)$
6. $3x+1=7(x-2)$
7. $6(2x-1)=5(x+7)$
8. $x-3(x+2)=4x-5$
9. $4x-(x-2)=6+3(1-x)$
10. $x(x-2)+2x(x-1)=x(3x+5)$
11. $5(x-3)-3(x+2)=7(x+4)-x-3$
12. $2\left[ x-3(x-5) \right]=5(x+1)$
13. $3-7x=4\left[ 3x-2(x+6) \right]$
14. $x-2\left[ 3x-4(5x-6) \right]=0$
15. $15-2x(x+1)=x(x+5)-3x(x+7)$
16. $2x-3\left[ x+2(x-3) \right]=7(x+1)$
17. $x(2x-1)-3(x+2)=5+2x(x+2)$
18. $x+3(x-4)=5-2(x-6)$
Solve each of the following equations for $x$ in terms of $a$.
19. $2x-3a=4(x-a)$
20. $3(x-2a)=7x+3$
21. $5(x+a-2)=3(a-2x)$
22. $a(x+2)=x(2a+3)$
23. $x(a+1)-a(x-3)=7$
24. $a(2x-3)+4(x-2)=5-3a$
Solve each of the following equations for $x$ in terms of any other variables appearing.
25. $12x+3p=5a$
26. $15xy+3z=4$
27. ${{b}^{2}}-4ax=0$
28. $a(x+3)=3x+7K$
29. $2a(x+b)=5ab+3bx$
30. $x(a-x)-b(a-x)=3ab-{{x}^{2}}$
Solve each of the following equations for $x$.
\begin{array}{l l}
31. \;\;\dfrac{2x}{3}+\dfrac{3x}{2}=2x+1\;\;&
32. \;\;\dfrac{2x}{3}-\dfrac{x}{5}=x-8\;\;\\
33. \;\;\dfrac{x+1}{7}-\dfrac{x-1}{5}=x-6\;\;&
34. \;\;\dfrac{x+1}{3}+\dfrac{2x+1}{5}=\dfrac{3x+2}{4}\;\;\\
35. \;\;\dfrac{x}{2}-\dfrac{x+1}{3}=\dfrac{x-3}{4}\;\;&
36. \;\;\dfrac{x+1}{7}-\dfrac{x-1}{2}=x+2\;\;\\
37. \;\;\dfrac{x+5}{7}+\dfrac{2x+1}{5}=2\;\;&
38. \;\;\dfrac{8-x}{7}+\dfrac{2-3x}{5}=6\;\;\\
39. \;\;\dfrac{x+2}{3}-\dfrac{x-2}{15}=\dfrac{12}{5}\;\;&
40. \;\;\dfrac{x+2}{2}+\dfrac{x-2}{5}=\dfrac{3}{4}\;\;\\
41. \;\;\dfrac{x+2}{2}-\dfrac{x-2}{10}=1\;\;&
42. \;\;\dfrac{x+2}{2}+\dfrac{x-2}{7}=\dfrac{1}{14}\;\; \\
\end{array}