8.1: Algebra Talk: Solving Equations by Seeing Structure
Find a solution to each equation mentally.
- $3 + x = 8$
- $10 = 12 - x$
- $x^2 = 49$
- $\frac13 x = 6$
Let's use diagrams to figure out which expressions are equivalent and which are just sometimes equal.
Find a solution to each equation mentally.
Here is a diagram of $x+2$ and $3x$ when $x$ is 4. Notice that the two diagrams are lined up on their left sides.
In each of your drawings below, line up the diagrams on one side.
Draw a diagram of $x+2$, and a separate diagram of $3x$, when $x$ is 3.
Draw a diagram of $x+2$, and a separate diagram of $3x$, when $x$ is 2.
Draw a diagram of $x+2$, and a separate diagram of $3x$, when $x$ is 1.
Draw a diagram of $x+2$, and a separate diagram of $3x$, when $x$ is 0.
Here is a list of expressions. Find any pairs of expressions that are equivalent. If you get stuck, try reasoning with diagrams.
$a+3$
$a+a+a$
$a \div \frac13$
$a \boldcdot 3$
$\frac13 a$
$3a$
$\frac{a}{3}$
$1a$
$a$
$3+a$
Below are four questions about equivalent expressions. For each one:
We can use diagrams showing lengths of rectangles to see when expressions are equal. For example, the expressions $x+9$ and $4x$ are equal when $x$ is 3, but are not equal for other values of $x$.
Sometimes two expressions are equal for only one particular value of their variable. Other times, they seem to be equal no matter what the value of the variable.
Expressions that are always equal for the same value of their variable are called equivalent expressions. However, it would be impossible to test every possible value of the variable. How can we know for sure that expressions are equivalent?
We use the meaning of operations and properties of operations to know that expressions are equivalent. Here are some examples:
In the coming lessons, we will see how another property, the distributive property, can show that expressions are equivalent.
Expressions that are always equal for the same value of their variable are called equivalent expressions.