Solving Linear Equations, All Mixed Up
This exercise mixes up problems from three earlier exercises:
- Solving Simple Linear Equations with Integer Coefficients
- Solving More Complicated Linear Equations with Integer Coefficients
- Solving Linear Equations involving Fractions
Examples
Solve:
$\,2x - 1 = 5\,$
Solution:
| $2x-1=5$ | original equation |
| $2x=6$ | add $\,1\,$ to both sides |
| $x = 3$ | divide both sides by $\,3$ |
Solve:
$3 - 2x = 5x + 1$
Solution:
| $3 - 2x = 5x + 1$ | original equation |
| $3 = 7x + 1$ | add $\,2x\,$ to both sides |
| $2 = 7x$ | subtract $\,1\,$ from both sides |
| $\frac{2}{7} = x$ | divide both sides by $\,7\,$ |
| $x = \frac{2}{7}$ | write in the most conventional way |
Solve:
$\displaystyle -3x -\frac{8}{9} = \frac{5}{6}$
Solution:
| $\displaystyle -3x -\frac{8}{9} = \frac{5}{6}$ | original equation |
| $\displaystyle 18\left(-3x -\frac{8}{9}\right) = 18(\frac{5}{6})$ | multiply both sides by $\,18\,,$ which is the least common multiple of $\,9\,$ and $\,6\,$ |
| $-54x - 16 = 15$ | simplify; all fractions are gone |
| $-54x = 31$ | add $\,16\,$ to both sides |
| $\displaystyle x = -\frac{31}{54}$ | divide both sides by $\,-54\,$ |
Practice
For more advanced students, a graph is available. For example, the equation $\,3 - 2x = 5x + 1\,$ is optionally accompanied by the graph of $\,y = 3 - 2x\,$ (the left side of the equation, dashed green) and the graph of $\,y = 5x + 1\,$ (the right side of the equation, solid purple).
Notice that you are finding the value of $\,x\,$ where these graphs intersect. Click the ‘Show/Hide Graph’ button to toggle the graph.