• Solve an equation with constants on both sides
  • Solve an equation with variables on both sides
  • Solve an equation with variables and constants on both sides

Assignment

Solve Equations with Constants on Both Sides

In 2.3 we formally move on to two-step equations, thought the text refers to them as having constants on both sides.

\[\begin{align} 7x + 8 = -13 \end{align}\]

Now that we have the basic mechanics as far as adding or multiplying, we’re going to start developing a strategy for solving these types of equations. The first step in that strategy is to move all the constants to the same side, specifically the side opposite the variables. In this example, we only have to divide after the move to determine the value of $x$.

\[\begin{align} 7x + 8 &= -13 \\ 7x + 8 - 8 &= -13 - 8 \\ 7x &= -21 \\ x &= -3 \end{align}\]

Don’t forget to check your work by plugging the value for $x$ back into the original equation.

\[\begin{align} 7(-3) + 8 &= -13 \\ -21 + 8 &= -13 \\ -13 &= -13 \end{align}\]

Solve Equations with Variables on Both Sides

The second step in the strategy is to get all the variable terms on the same side. Again, it should be the side opposite the constants. You can add or subtract variable terms just like constants.

\[\begin{align} 9x &= 8x -6 \\ 9x - 8x &= 8x -6 - 8x \\ x &= -6 \end{align}\]

Solve Equations with Variables and Constants on Both Sides

When you have constants and variables on both sides, we just do both the steps above. The order doesn’t matter much, but make sure you first decide which side will house your constants and which your variables.

\[\begin{align} 8x - 4 &= -2x + 6 \\ 8x - 4 + 4&= -2x + 6 + 4\\ 8x &= -2x + 10 \\ 8x + 2x &= -2x + 10 + 2x\\ 10x &= 10 \\ x &= 1 \end{align}\]