• Solve equations using a general strategy
  • Classify equations

Assignment

Solve Equations Using the General Strategy

We now have everything we need solve linear equations and can form a list of steps to follow.

General strategy for solving linear equations

  1. Simplify Use the Distributive Property to remove any parentheses and combine like terms on both sides of the equation.
  2. Collect Use the Addition and Subtraction Properties of Equality to collect all the variable terms on one side, and constants on the other.
  3. Solve Make the coefficient of the variable term to equal to 1 by using the Multiplication or Division Property of Equality.
  4. Check Substitute the solution into the original equation to make sure the result is a true statement.
\[\begin{align} 5(x - 3) - 5 &= -10 \\ (5x - 15) - 5 &= -10 && \text{Simplify} \\ 5x - 20 &= -10 && \text{Simplify again} \\ 5x - 20 + 20 &= -10 + 20 && \text{Collect the constants} \\ 5x &= 10 \\ x &= 2 &&\text{Solve} \\[1em] 5\left((2) - 3\right) - 5 &= -10 \\ 5(-1) - 5 &= -10 \\ -10 &= -10 &&\text{Check} \end{align}\]

Classify Equations

Every equation we have looked at so far is considered a conditional equation. They are true statements if you place the correct value (or values) in for the variables. If you place the incorrect value, then the equation is false.

The equation ${x+3=6}$ is a conditional equation because it’s only true when $x=3$. Any other value results in a false statement.

An identity is always true, no matter what the value used for the variable. The simplest version of an identity is $x=x$, but they can be appear in other forms.

\[\begin{align} 2x + 6 &= 2(x + 3) \label{ident}\\ 2x + 6 &= 2x + 6 \end{align}\]

We can continue to follow our strategy, since both sides are exactly the same we can conclude that equation $\ref{ident}$ is an identity.

The last type of equation we’ll look at is a contradiction. These are false no matter what, for examples ${x = x + 1}$. Like identities, these can be hidden until a few steps into your solving strategy.