# Solving Linear Equations, All Mixed Up

This exercise mixes up problems from three earlier exercises:

## 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.