Linear equations

Chapter 1 · 1 lesson · 64 questions

Unit 1: Solving a linear equation, or the expression it asks for

Whatever you do to one side of an equation, do to the other side too.
First multiply out any parentheses: multiply every term inside, sign included.
Then collect the $x$ -terms on one side and the numbers on the other.
If the question asks for an expression, such as $3x + 2$ , find that whole expression. You may not need $x$ itself.

Example

$5(x - 2)$ $= 2x + 11$

What value of $x$ is the solution to the given equation?

The plan

Desmos: A short equation is just as quick by hand. Desmos is a good check: type the equation and click where the vertical line meets the $x$ -axis.

Step 1: Simplify the parentheses
- Multiply each term inside the parentheses by $5$ .
- $5 \cdot x = 5x$ and $5 \cdot (-2) = -10$ .
- The equation is now $5x - 10$ $= 2x + 11$ .
Step 2: Isolate the variable
- Subtract $2x$ from both sides: $3x - 10 = 11$ .
- Add $10$ to both sides: $3x$ $= 11 + 10$ $= 21$ .
- Divide both sides by $3$ : $x = \dfrac{21}{3} = 7$ .
- Check the left side: $5(7 - 2)$ $= 5 \cdot 5$ $= 25$ .
- Check the right side: $2 \cdot 7 + 11$ $= 14 + 11$ $= 25$ . The two sides match.

The correct answer is

7

Made with Desmos

If $2(3x + 2) - 5$ $= 19$ , what is the value of $3x + 2$ ?

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Answer

The correct answer is $12$ .

Step 1: Isolate the expression
- Treat $3x + 2$ as one unit. Do not multiply it out.
- Add $5$ to both sides: $2(3x + 2)$ $= 19 + 5$ $= 24$ .
- Divide both sides by $2$ : $3x + 2 = \dfrac{24}{2} = 12$ .

Multiplying only the first term. $5(x - 2)$ is $5x - 10$ , not $5x - 2$ . With $5x - 2 = 2x + 11$ you get $3x = 13$ and $x = \dfrac{13}{3}$ , a wrong answer.
Answering with $x$ when an expression is asked for. In Your turn, $x = \dfrac{10}{3}$ , but the question asks for $3x + 2$ , which is $12$ .

Typing tips for Desmos

In the keys lines, → means the right-arrow key.
Typing / starts a fraction, and everything after it goes into the denominator until you press →. So 42/120*100 typed straight through becomes 42 ÷ (120 × 100). Type 42/120 → *100 instead.
Desmos solves equations in x (and y) only. An equation in t, such as 3t + 7 = 22, gives an error until you retype it with x.