First-degree equations can be solved by applying different operations to both sides of the equal sign so that we simplify the equation and solve for the variable.

In this article, we will look at a summary of solving first-degree equations. In addition, we will look at several examples with answers and practice problems.

## Summary of first-degree equations

Recall that first-degree equations are equations in which all variables have a maximum exponent of 1. For example, the equations $latex 3x+2=6$ and $latex 4x+3=2x-1$ are first degree equations.

To solve first-degree equations, we remember that we can apply any operation to the equation as long as we perform that operation on both sides of the equal sign. We can follow the following steps to solve first-degree equations:

**Step 1:** Remove grouping signs and combine like terms.

**Step 2:** Isolate the terms with variables on one side of the equation.

**Step 3:** Use multiplication and division to solve for the variable entirely and to find its value.

## First-degree equations – Examples with answers

**EXAMPLE 1**

Solve the equation $latex 4x+2=10$.

##### Solution

**Step 1:** Simplify: We do not have grouping signs or like terms.

**Step 2: **Solve for the variable: We use subtraction to solve for the variable:

$latex 4x+2=10$

$latex 4x+2-2=10-2$

$latex 4x=8$

**Step 3:** Solve: We use division to solve:

$latex 4x=8$

$$\frac{4}{4}x=\frac{8}{4}$$

$latex x=2$

**EXAMPLE 2**

Find the value of *x* in the equation $latex 5x-6=15-2x$.

##### Solution

**Step 1:** Simplify: We do not have grouping signs or like terms.

**Step 2: **Isolate the variable: We use sums to solve for the variable:

$latex 5x-6=15-2x$

$latex 5x-6+6=15-2x+6$

$latex 5x=21-2x$

$latex 5x+2x=21-2x+2x$

$latex 7x=21$

**Step 3:** Solve: We divide by 7 to solve:

$latex 7x=21$

$$\frac{7}{7}x=\frac{21}{7}$$

$latex x=3$

**EXAMPLE 3**

Find the value of *x* in the equation $latex 4(2x+4)=-8$.

##### Solution

**Step 1:** Simplify: We expand the parentheses:

$latex 4(2x+4)=-8$

$latex 8x+16=-8$

**Step 2: **Isolate the variable: We use subtraction to solve for the variable:

$latex 8x+16=-8$

$latex 8x+16-16=-8-16$

$latex 8x=-24$

**Step 3:** Solve: We divide by 8 to solve:

$latex 8x=-24$

$$\frac{8}{8}x=\frac{-24}{8}$$

$latex x=-3$

**EXAMPLE 4**

Solve the equation $latex 2(4x-11)=3(2x-4)$.

##### Solution

**Step 1:** Simplify: We expand the parentheses on both sides:

$latex 2(4x-11)=3(2x-4)$

$latex 8x-22=6x-12$

**Step 2: **Isolate the variable: We use addition and subtraction to solve for the variable:

$latex 8x-22+22=6x-12+22$

$latex 8x=6x+10$

$latex 8x-6x=6x+10-6x$

$latex 2x=10$

**Step 3:** Solve: We divide by 2 to solve:

$latex 2x=10$

$$\frac{2}{2}x=\frac{10}{2}$$

$latex x=5$

**EXAMPLE 5**

Solve the equation $latex 5(x-2)-3=4(2x+3)-4x$.

##### Solution

**Step 1:** Simplify: We expand the parentheses on both sides and combine like terms:

$latex 5(x-2)-3=4(2x+3)-4x$

$latex 5x-10-3=8x+12-4x$

$latex 5x-13=4x+12$

**Step 2: **Isolate the variable: We solve using addition and subtraction:

$latex 5x-13+13=4x+12+13$

$latex 5x=4x+25$

$latex 5x-4x=4x+25-4x$

$latex x=25$

**Step 3:** Solve: In this case, we no longer have to divide:

$latex x=25$

**EXAMPLE 6**

Solve the equation $latex \frac{5-2x}{3}=\frac{-x+7}{6}$.

##### Solution

**Step 1:** Simplify: We multiply the entire equation by 6 to eliminate the fractions:

$$\frac{5-2x}{3}=\frac{-x+7}{6}$$

$latex 2(5-2x)=-x+7$

$latex 10-4x=-x+7$

**Step 2: **Isolate the variable: We solve using addition and subtraction:

$latex 10-4x-10=-x+7-10$

$latex -4x=-x-3$

$latex -4x+x=-x-3+x$

$latex -3x=-3$

**Step 3:** Solve: We divide by -3 to solve:

$$\frac{-3}{-3}x={-3}{-3}$$

$latex x=1$

**EXAMPLE 7**

Solve the equation $latex 4(2x-5)=\frac{x-1}{2}+3$.

##### Solution

**Step 1:** Simplify: We multiply the entire equation by 2 to eliminate the fractions. Then we remove the parentheses and combine like terms:

$$4(2x-5)=\frac{x-1}{2}+3$$

$latex 8(2x-5)=x-1+6$

$latex 16x-40=x+5$

**Step 2: **Isolate the variable: We solve using addition and subtraction:

$latex 16x-40+40=x+5+40$

$latex 16x=x+45$

$latex 16x-x=x+45-x$

$latex 15x=45$

**Step 3:** Solve: We divide by 15 to solve:

$$\frac{15}{15}x=\frac{45}{15}$$

$latex x=3$

**EXAMPLE **8

Find the value of *x* in the equation: $latex \frac{4x-9}{3}+2=3(x-2)$.

##### Solution

**Step 1:** Simplify: We can multiply both sides of the equation by 3. Also, we expand the parentheses and combine like terms:

$$\frac{4x-9}{3}+2=3(x-2)$$

$latex 4x-9+6=9(x-2)$

$latex 4x-3=9x-18$

** Step 2:** Isolate the variable: We add 3 and subtract 9

*x*from both sides:

$latex 4x-3+3=9x-18+3$

$latex 4x=9x-15$

$latex 4x-9x=9x-15-9x$

$latex -5x=-15$

**Step 3:** Solve: We divide both sides by -5:

$$\frac{-5x}{-5}=\frac{-15}{-5}$$

$latex x=3$

**EXAMPLE **9

**EXAMPLE**

Solve the equation $latex 10(2w-5)=2w+2(w+1)$ and find the value of *w*.

##### Solution

**Step 1:** Simplify: We just need to expand the parentheses and combine like terms:

$latex 10(2w-5)=2w+2(w+1)$

$latex 20w-50=2w+2w+2$

$latex 20w-50=4w+2$

**Step 2:** Isolate the variable: We add 50 and subtract 4*w* from both sides:

$latex 20w-50=4w+2$

$latex 20w-50+50=4w+2+50$

$latex 20w=4w+52$

$latex 20w-4w=4w+52-4w$

$latex 16w=52$

**Step 3:** Solve: We divide both sides by 16 and simplify the fraction:

$$\frac{16w}{16}=\frac{52}{16}$$

$latex x=\frac{13}{4}$

**EXAMPLE **10

**EXAMPLE**10

Solve the equation $latex 2\left( \frac{x+2}{4}\right)+2=\frac{3x}{4}+2$ and find the value of *x*.

##### Solution

**Step 1:** Simplify: We multiply the 2 in the fraction on the left-hand side to simplify it, then we multiply the entire equation by 4 to remove the fractions and combine like terms:

$$2\left( \frac{x+2}{4}\right)+2=\frac{3x}{4}+2$$

$$\frac{x+2}{2}+2=\frac{3x}{4}+2$$

$latex 2(x+2)+8=3x+8$

$latex 2x+4+8=3x+8$

$latex 2x+12=3x+8$

**Step 2:** Isolate the variable: We subtract 12 and 3*x* from both sides:

$latex 2x+12-12=3x+8-12$

$latex 2x=3x-4$

$latex 2x-3x=3x-4-3x$

$latex -x=-4$

**Step 3:** Solve: We divide both sides by -1:

$$\frac{-x}{-1}=\frac{-4}{-1}$$

$latex x=4$

## First-degree equations – Practice problems

#### Find the value of t en la equation $latex \frac{x+3}{2}=\frac{x+6}{4}+\frac{1}{2}$.

Write the answer in the input box.

## See also

Interested in learning more about solving equations? Take a look at these pages:

### Learn mathematics with our additional resources in different topics

**LEARN MORE**