Solve Linear Equations and Justify Each Step | Geometry | LevelKite

An equation is a statement that two expressions have the same value. When it contains a variable, it can be true for some values and false for others. A solution is a value of the variable that makes the equation true. For example, $2$ is a solution of $x + 3 = 5$ because $2 + 3 = 5$ , but $4$ is not, because $4 + 3 = 7$ .

A linear equation in $x$ is an equation in which each side simplifies to the form $ax + b$ , where $a$ and $b$ are real numbers. So $x$ never appears squared or in a denominator. The number $a$ is the coefficient of $x$ : the number that multiplies it.

In geometry, unknown lengths and angle measures often lead to linear equations. In this lesson, you will solve them and give a reason for every step.

Key concept

Keep the two sides equal

To solve an equation means to find all of its solutions. You do this by replacing it with simpler equivalent equations, meaning equations with the same solutions, until you can read the answer. Picture a balanced scale: you keep it balanced by doing the same operation to both entire sides. You may add or subtract any real number or variable term, or multiply or divide by a nonzero number. Division by zero is undefined, so $0$ can never be your divisor.

Properties of equality: accepted facts you can cite as reasons

Property of Equality	What it says	Condition
Addition	If $a = b$ , then $a + c = b + c$ .	$c$ is any real number. A term such as $3x$ counts, since it is a real number for each $x$ .
Subtraction	If $a = b$ , then $a - c = b - c$ .	Same as for addition.
Multiplication	If $a = b$ , then $ac = bc$ .	When solving, use $c \neq 0$ . Using $c = 0$ gives $0 = 0$ and loses all information about $x$ .
Division	If $a = b$ and $c \neq 0$ , then $\frac{a}{c} = \frac{b}{c}$ .	$c \neq 0$ , because division by zero is undefined.
Symmetric	If $a = b$ , then $b = a$ .	None.
Substitution	If $a = b$ , then $b$ can replace $a$ in any expression.	None.

A two-step equation is one you can solve with two inverse operations. An inverse operation undoes another operation: addition and subtraction undo each other, and so do multiplying and dividing by the same nonzero number.

To choose the operations, read what the equation does to $x$ . In $3x + 5 = 26$ , the left side starts with $x$ , multiplies by $3$ , then adds $5$ . To get back to $x$ , undo those operations in reverse order, just as you take off your shoes before your socks.

\text{Forward: } x \xrightarrow{\times 3} 3x \xrightarrow{+5} 3x + 5 \qquad \text{Backward: } 26 \xrightarrow{-5} 21 \xrightarrow{\div 3} 7

Worked example

A two-step equation

Given $3x + 5 = 26$ , find the value of $x$ .

Adding 5 was the last operation on x, so undo it first. Subtract 5 from both sides. Reason: Subtraction Property of Equality.
$3x + 5 - 5 = 26 - 5$
Simplify each side. On the left, 5 - 5 = 0. On the right, 26 - 5 = 21.
$3x = 21$
Now undo the multiplication by 3. Divide both sides by 3. Reason: Division Property of Equality, which applies because 3 is not 0.
$\frac{3x}{3} = \frac{21}{3}$
Simplify each side. On the left, 3 divided by 3 is 1, which leaves x. On the right, 21 divided by 3 is 7.
$x = 7$
Check with the Substitution Property of Equality. If x = 7, then 7 can replace x. Simplify the original left side by itself. The result, 26, matches the right side.
$3(7) + 5 = 21 + 5 = 26$

Answer

The solution of $3x + 5 = 26$ is $x = 7$ .

Notice the two directions in that work. The solving steps show that if $3x + 5 = 26$ , then $x = 7$ , so $7$ is the only possible solution. The check shows the converse, the statement with the if and then parts swapped: if $x = 7$ , then $3x + 5 = 26$ . So $7$ really is a solution.

This is why you compute each side on its own during a check. If you started by assuming the sides are equal, you would be using the conclusion as its own reason.

The same method with other operations

Equation	Undo the last operation	Undo the first operation	Check by substitution
$\frac{x}{4} - 3 = 5$	Add $3$ to both sides (Addition Property of Equality): $\frac{x}{4} = 8$	Multiply both sides by $4$ , which is not $0$ (Multiplication Property of Equality): $x = 32$	$\frac{32}{4} - 3 = 8 - 3 = 5$
$-2x + 9 = 1$	Subtract $9$ from both sides (Subtraction Property of Equality): $-2x = -8$	Divide both sides by $-2$ , which is not $0$ (Division Property of Equality): $x = 4$	$-2(4) + 9 = -8 + 9 = 1$

A multi-step equation takes more than two operations to solve, often because a side needs simplifying first. Begin by using what you already know on each side separately: the Distributive Property, then combining like terms. Simplifying a side does not change its value, so the result is an equivalent equation. In the example below, simplifying leaves a two-step equation, and you know how to finish from there.

Worked example

A multi-step equation

Given $4(x - 3) + 2x = 18$ , find the value of $x$ .

Apply the Distributive Property on the left: multiply both x and -3 by 4. The value of the side does not change.
$4x - 12 + 2x = 18$
Combine the like terms 4x and 2x.
$6x - 12 = 18$
Add 12 to both sides to undo the subtraction. Reason: Addition Property of Equality.
$6x - 12 + 12 = 18 + 12 \;\Rightarrow\; 6x = 30$
Divide both sides by 6. Reason: Division Property of Equality, which applies because 6 is not 0.
$\frac{6x}{6} = \frac{30}{6} \;\Rightarrow\; x = 5$
Check by substitution in the original equation, since a mistake in an early step would carry into every later equation. The left side equals 18, which matches the right side.
$4(5 - 3) + 2(5) = 4(2) + 10 = 8 + 10 = 18$

Answer

The solution of $4(x - 3) + 2x = 18$ is $x = 5$ .

Sometimes $x$ appears on both sides, as in $7x - 4 = 3x + 20$ . Now the value of each side depends on $x$ , and a solution is a value of $x$ that gives both sides the same value. The table below uses substitution to compare the sides, and they match at $x = 6$ .

Trying values is slow, it can miss a solution that is not a whole number, and it does not show that no other solution exists. Instead, use the properties of equality to collect the variable terms on one side and the numbers on the other. That usually leaves an equation you can finish with inverse operations, as before.

Each side of the equation at four values of x

Value of x	Left side, 7x - 4	Right side, 3x + 20	Same value?
$4$	$24$	$32$	No
$5$	$31$	$35$	No
$6$	$38$	$38$	Yes
$7$	$45$	$41$	No

Worked example

Variables on both sides

Given $7x - 4 = 3x + 20$ , find the value of $x$ .

Collect the variable terms on the left. Subtract 3x from both sides (Subtraction Property of Equality), then combine like terms.
$7x - 4 - 3x = 3x + 20 - 3x \;\Rightarrow\; 4x - 4 = 20$
Collect the numbers on the right. Add 4 to both sides (Addition Property of Equality).
$4x - 4 + 4 = 20 + 4 \;\Rightarrow\; 4x = 24$
Divide both sides by 4, which is not 0 (Division Property of Equality).
$\frac{4x}{4} = \frac{24}{4} \;\Rightarrow\; x = 6$
Check by substitution in the original equation, simplifying each side separately. Both sides equal 38.
$7(6) - 4 = 42 - 4 = 38 \quad\text{and}\quad 3(6) + 20 = 18 + 20 = 38$

Answer

The solution of $7x - 4 = 3x + 20$ is $x = 6$ , and the algebra shows it is the only solution.

The equations so far had no geometric meaning, so the only condition to test was a nonzero divisor. In geometry, a solution must also give measures that can exist: a length must be positive, and an angle measure must be between $0^\circ$ and $180^\circ$ . Test the measure, not $x$ itself. The value of $x$ may be negative as long as the measure is allowed.

Substitution alone does not test these conditions. Suppose each side of $2x + 10 = x + 4$ is given as a segment length. By the Subtraction Property of Equality, subtracting $x$ from both sides gives $x + 10 = 4$ , and subtracting $10$ from both sides gives $x = -6$ . Substitution gives $2(-6) + 10 = -2$ on the left and $-6 + 4 = -2$ on the right, so $-6$ solves the equation. A length cannot be $-2$ , though, so no such segment exists.

Worked example

Variables on both sides with an angle condition

Given $2(3x - 5) + 4 = 9x - 36$ , where each side is an angle measure in degrees, find $x$ and the measure.

Simplify the left side: apply the Distributive Property, then combine like terms.
$6x - 10 + 4 = 9x - 36 \;\Rightarrow\; 6x - 6 = 9x - 36$
Subtract 6x from both sides (Subtraction Property of Equality). Removing the smaller variable term keeps the remaining coefficient positive.
$6x - 6 - 6x = 9x - 36 - 6x \;\Rightarrow\; -6 = 3x - 36$
Add 36 to both sides (Addition Property of Equality).
$-6 + 36 = 3x - 36 + 36 \;\Rightarrow\; 30 = 3x$
Divide both sides by 3, which is not 0 (Division Property of Equality). Then write x on the left (Symmetric Property of Equality).
$\frac{30}{3} = \frac{3x}{3} \;\Rightarrow\; 10 = x \;\Rightarrow\; x = 10$
Check by substitution in the original equation, simplifying each side separately. Both sides equal 54.
$2(3 \cdot 10 - 5) + 4 = 2(25) + 4 = 54 \quad\text{and}\quad 9(10) - 36 = 90 - 36 = 54$
Test the geometric condition. An angle measure must be between 0 and 180 degrees, and 54 is, so the value is allowed.
$0 < 54 < 180$

Answer

The solution is $x = 10$ , and each side gives an angle measure of $54^\circ$ .

Solving a linear equation means building a chain of equivalent equations, with a named reason for each link. When needed, simplify each side and collect the variable terms on one side. Then undo the remaining operations in reverse order. At every step, do the same operation to both sides, and divide only by a nonzero number. Finally, check your value in the original equation and test any measure it gives.

Next, you will rearrange formulas to solve for a variable, and you will write and solve equations from geometric descriptions.