Algebra 1 Two Step Equations

So, weve made it past one-footstep equations! Woo-hoo!

Our reward? Solving ii-step equations!

Dont worry: whether this is your get-go experience with two-step equations, or you lot are reviewing for an exam, this blog volition guide you through defining two-step equations, examples of two-footstep equations, and how to solve two-step equations (including fractions and discussion issues). Lets become started!

What is a two-step equation?
Examples of two-footstep equations
How to solve two-stride equations
- Example 1
- Example 2
How to solve two-step equations with fractions
Ii-step equation word problems
- Example 1
- Example two
Solving Two-Step Equations: Keys to Remember

What is a 2-step equation?

Remember, an equation is a mathematical sentence that uses an equal sign, $=$ , to show that two expressions are equal.

Very like to ane-step equations, a two-pace equation is an equation that only requires 2 steps to solve. Nosotros will use a mix of improver, subtraction, multiplication, and division to solve these equations.

Examples of two-step equations

Two-footstep equations come up in many types. Yous might take some equations that require subtraction, then partitioning to solve, or an equation that requires multiplication, then division to solve.

Here are some examples of two-stride equations:

$4 + 3x = 16$	$\dfrac{y}{2} - 9 = 11$
$-8 - 2m = 12$	$\dfrac{4x}{three} = 8$

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How to solve two-stride equations

Solving two-step equations on a whiteboard

Remember: To solve equations, nosotros must use inverse operations to isolate the variable. Examples of inverse operations are:

$\text{Addition} \leftrightarrow \text{Subtraction}$

$\text{Multiplication} \leftrightarrow \text{Partition}$

We must first eliminate any constants from the side of the equation with the variable. Additionally, whatsoever we do to i side of the equation, nosotros must also do to the other. Here are two examples of how to solve 2-step equations:

Example 1

Commencement, lets solve for $10$ in the post-obit equation:

$5x - 4 = sixteen$	Original Equation
$5x - 4 \textcolor{red}{+ 4} = xvi \color{red}{+ iv }$	Add together $4$ to both sides
$5x = 20$	Simplify
$\dfrac{5x}{\textcolor{crimson}{five}} = \dfrac{twenty}{\textcolor{red}{v}}$	Divide Each Side past $5$
$ten = four$	Simplify

To check your answer, y'all tin can substitute $four$ into the variable to meet if the equation is truthful:

$5x - 4 = 16$

$v(4) - 4 = 16$

$xx - four = 16$

$16 = 16 \checkmark$

Thus, $x = 4$ is the correct solution.

Example ii

Now, we tin can trying solving for $y$ in the following equation:

$\dfrac{y}{vii} + 3 = 4$	Original Equation
$\dfrac{y}{7} + three \textcolor{red}{- iii} = 4 \textcolor{red}{- iii}$	Decrease $three$ from each side
$\dfrac{y}{7} = 1$	Simplify
$\dfrac{y}{7} \textcolor{red}{\cdot seven} = 1 \textcolor{scarlet}{\cdot 7 }$	Multiply each side by $7$
$y = vii$	Simplify

To check you answer, you tin can simplify substitute $vii$ into the variable to see if the equation is true:

$\dfrac{y}{seven} + iii = 4$

$\dfrac{seven}{7} + 3 = 4$

$i + 3 = 4$

$4 = 4 \checkmark$

Thus, $y = 7$ is the right solution.

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For more, watch the video from mathantics beneath showing how the solve two-pace equations:

How to solve ii-step equations with fractions

Unfortunately, equations do non always comprise only whole numbers. Never fearfulness! We can still solve two-step equations even when fractions are involved.

Here is an example of solving a two-step equation with a fraction:

Solve for $g$ in the following equation:

$\dfrac{ii}{3}1000 + half dozen = 12$	Original Equation
$\dfrac{2}{three}m + half dozen \textcolor{red}{- 6} = 12 \textcolor{crimson}{ - half dozen}$	Subtract $vi$ from each side
$\dfrac{2}{3}m = 6$	Simplify
$\textcolor{crimson}{ \dfrac{3}{2}} \cdot \dfrac{2}{3}m = \textcolor{blood-red}{\dfrac{3}{two}} \cdot 6$	Multiply each side by $\dfrac{3}{ii}$
$m = \dfrac{18}{2}$	Simplify
$one thousand = nine$	Simplify

To check you answer, yous can simplify substitute $9$ into the variable to come across if the equation is truthful:

$\dfrac{2}{3}thousand + six = 12$

$\dfrac{2}{iii} \cdot (9) + 6 = 12$

$\dfrac{18}{3} + 6 = 12$

$6 + 6 = 12$

$12 = 12 \checkmark$

Thus, $m = 9$ is the correct solution.

Is there a way to brand solving two-footstep equations with fractions easier? Im glad y'all asked! If you desire to eliminate fractions completely when solving a two-step equation, you lot can simply multiply the whole equation by the Least Mutual Denominator. Here is an example showing this method:

Solve for $x$ in the post-obit equation:

\dfrac{ten}{2} - \dfrac{1}{3} = \dfrac{one}{half dozen}

Original Equation

Since the denominators are $2, 3, \text{and } 6$ , the to the lowest degree common denominator would exist $6$ . Therefore, to eliminate all fractions from the problem, we would multiply each term past $half-dozen$ .

$\textcolor{red}{six \cdot} \dfrac{ten}{2} - \textcolor{red}{6 \cdot} \dfrac{i}{iii} = \textcolor{red}{vi \cdot} \dfrac{1}{6}$	Multiply each term by $6$
$\dfrac{6x}{2} - \dfrac{six}{3} = \dfrac{six}{6}$	Simplify
$3x - two = 1$	Simplify
$3x - ii \textcolor{red}{+ 2} = ane \textcolor{red}{+ two}$	Add $2$ to each side
$3x = three$	Simplify
$\dfrac{3x}{\textcolor{red}{3}} = \dfrac{3}{\textcolor{red}{3}}$	Divide each side by $iii$
$ten = ane$	Simplify

To check you respond, you can simplify substitute $1$ into the variable to meet if the equation is true:

$\dfrac{x}{2} - \dfrac{1}{3} = \dfrac{ane}{6}$

$\dfrac{ane}{two} - \dfrac{1}{3} = \dfrac{i}{6}$

$\dfrac{3}{6} - \dfrac{2}{6} = \dfrac{ane}{half-dozen}$

$\dfrac{i}{6} = \dfrac{1}{6} \checkmark$

Heres a video from Brian McLogan on how to solve ii-step equations with fractions:

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Two-footstep equation word issues

Similar to One-Footstep Equations, we tin model real-life scenarios with two-step equations. One time we model the situation with an equation, nosotros simply solve as we have above.

For instance, model the following situations with an equation and discover a solution that makes the situation true.

Case 1

Toms brother buys three tickets to a baseball merely does non tell Tom how much each ticket costs. Tom actually wants to reimburse his brother for his ticket, then he wants to notice the cost of one ticket. He knows the total toll of all the tickets was

\$250

. He also knows the website where his brother bought the tickets charges a flat

\$10

processing fee.

Create an equation that models the state of affairs and solve the equation to find, $c$ , the cost of i ticket.

Solution: To model the following situation, nosotros will create an equation to prove the total cost of the tickets. We will represent the toll of ane ticket with the variable, $c$

$3c + 10 = 250$

To solve for $c$ , nosotros will get-go decrease $10$ from each side:

$3c + 10 \textcolor{carmine}{- 10} = 250 \textcolor{red}{- ten}$

$3c = 240$

So to find the cost of i ticket, we will dissever each side by $iii$

$c = 80$

Therefore, the cost of one ticket is $\$80$

Example ii

Four boutiques decide to host a Fall Style Market. These boutiques will split the profits from the event evenly. The four boutiques did have to pay for a portable toilet for the consequence, which price $50 and will exist paid for out of the profits. After calculating the profits and paying for the portable toilet, each boutique took home

\$400

. How much total profits did the consequence make?

Create an equation that models the state of affairs and solve the equation to observe, $t$ , the total profit made at the issue.

Solution: To model the following situation, we will create an equation to show the total profit from the result. We will represent the full profit made at the event with the variable, $t$

$\dfrac{(t - l)}{iv} = 400$

To solve for $t$ , we must first multiply each side past $4$ to eliminate the denominator

$\textcolor{cerise}{4 \cdot} \dfrac{(t - 50)}{four} = \textcolor{carmine}{4 \cdot} 400$

$t - 50 = 1600$

$t - 50 \textcolor{red}{+ 50} = 1600 \textcolor{red}{+ 50}$

$t=1650$

Therefore, the total profits made past the 4 boutiques was $\$1650$

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Solving Two-Stride Equations: Keys to Remember

Young African American man smiling and looking at empty space while sitting on chair and using laptop against blue background

Remember, merely like solving Ane-Pace Equations there are some central facts to call back:

A ii-step equation is an equation that requires two steps to solve
Nosotros must eliminate any constant that is on the same side as the variable outset
To solve, use the changed operations to isolate the variable by itself
Call up whatsoever you do to one side, y'all must practice to the other
To check the solution, simply substitute the value into the variable to see if the equation is truthful
You tin model real-life situations with an equation and solve for a correct solution

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Read these other helpful posts:

Solving One-Footstep Equations
Solving Multi-Pace Equations
Forms of Linear Equations
View ALL Algebra 1 Review Guides