Solving Two-Step Equations Using Distributive Property

Unlock the power of distributive property in solving two-step equations. Our comprehensive guide offers clear explanations, step-by-step solutions, and practice problems to boost your algebra proficiency.

Solve the equation using model. $4\left( {x + 1} \right) = 12$

Step 1: Understand the Problem

To solve the equation $4(x + 1) = 12$ using a model, we need to understand what the equation represents. The equation tells us that we have four groups of $x + 1$ and that these groups together equal 12. Our goal is to find the value of $x$.

Step 2: Represent the Equation Visually

We start by representing the left side of the equation visually. We have four groups of $x + 1$. This means we need to draw four sets of $x$ tiles and four sets of +1 tiles. Each group looks like this: $x + 1$.

So, we draw four groups:

• Group 1: $x + 1$
• Group 2: $x + 1$
• Group 3: $x + 1$
• Group 4: $x + 1$

On the right side of the equation, we have 12 positive units. We represent this by drawing 12 positive tiles.

Step 3: Simplify the Visual Representation

Next, we simplify the visual representation. We combine all the $x$ tiles and all the +1 tiles on the left side. This gives us:

• 4 $x$ tiles
• 4 +1 tiles

So, the left side of the equation is now represented as $4x + 4$. The right side remains as 12 positive tiles.

Step 4: Isolate the Variable

To isolate the variable $x$, we need to get rid of the +4 on the left side. We do this by adding -4 (negative 4) to both sides of the equation. This step ensures that we maintain the equality of the equation.

On the left side, adding -4 cancels out the +4, leaving us with just $4x$. On the right side, we subtract 4 from 12, which gives us 8 positive tiles.

So, the equation now looks like this: $4x = 8$.

Step 5: Solve for $x$

Now, we need to find the value of $x$. We have 4 $x$ tiles that equal 8 positive tiles. To find the value of one $x$ tile, we divide the 8 positive tiles into 4 equal groups.

Each group will have 2 positive tiles. Therefore, each $x$ tile is worth 2 positive tiles.

So, $x = 2$.

Conclusion

By using a visual model, we have successfully solved the equation $4(x + 1) = 12$. The value of $x$ is 2. This method helps in understanding the distributive property and the steps involved in isolating the variable to find its value.

Here are some frequently asked questions about solving two-step linear equations using the distributive property:

1. What is the distributive property and how does it apply to solving equations?

The distributive property states that a(b + c) = ab + ac. In equation solving, it's used to simplify expressions within parentheses. For example, in 3(x + 2) = 15, we distribute the 3 to get 3x + 6 = 15, making the equation easier to solve.

2. Why is it important to apply the distributive property before solving the equation?

Applying the distributive property first simplifies the equation, converting it into a standard two-step linear equation. This makes the subsequent steps of isolating the variable and solving much more straightforward.

3. How do you handle negative numbers when using the distributive property?

When distributing a negative number, remember to change the signs of all terms inside the parentheses. For example, -2(x - 3) becomes -2x + 6, not -2x - 6. Pay extra attention to these sign changes to avoid errors.

4. Can the distributive property be used with variables other than x?

Yes, the distributive property works with any variable. Whether it's x, y, z, or any other letter, the principle remains the same. For instance, 4(y + 1) = 4y + 4, regardless of the variable used.

5. How can I check if I've applied the distributive property correctly?

After applying the distributive property, you can check your work by expanding the original expression and comparing it to your result. For example, if you distributed 2(x + 3) to get 2x + 6, you can verify this by multiplying 2 by x and 2 by 3 separately.

Understanding the foundation of algebra is crucial when tackling more complex problems like solving two-step linear equations using the distributive property: a(x + b) = c. To master this concept, it's essential to grasp several prerequisite topics that build the necessary skills and knowledge.

One of the fundamental skills required is simplifying algebraic expressions. This ability allows you to manipulate and simplify the equation a(x + b) = c, making it easier to solve. Similarly, combining like terms is crucial when working with distributive property, as it helps in organizing and simplifying the equation after expansion.

When dealing with two-step equations, you'll often encounter solving equations with variables on both sides. This skill is particularly relevant as it prepares you for more complex equation structures and teaches you how to balance equations effectively.

Isolating variables in equations is another critical skill that directly applies to solving two-step linear equations. It teaches you how to manipulate the equation to get the variable by itself, which is a key step in the solving process.

Understanding the concept of applying inverse operations in algebra is vital when solving equations. This principle is used extensively in two-step equations to isolate the variable and find its value.

Another important aspect is understanding negative signs in equations. This skill is crucial when dealing with the distributive property, as it affects how terms are combined and simplified.

Lastly, proficiency in solving equations with fractions and decimals is beneficial, as these types of numbers often appear in real-world applications of two-step linear equations.

By mastering these prerequisite topics, you'll build a strong foundation for tackling more advanced concepts like solving two-step linear equations using the distributive property. Each of these skills contributes to your overall understanding and ability to approach complex algebraic problems with confidence. Remember, algebra is a cumulative subject, and each new concept builds upon previous knowledge. So, take the time to thoroughly understand these prerequisites, and you'll find yourself well-prepared for the challenges ahead in your algebraic journey.

• Model and solve one-step linear equations: ax = b, x/a = b
• Solving two-step linear equations using addition and subtraction: ax + b = c
• Solving two-step linear equations using multiplication and division: x/a + b = c

• Solving linear equations using multiplication and division
• Solving two-step linear equations: ax + b = c, x/a + b = c
• Solving linear equations using distributive property: a(x + b) = c
• Solving linear equations with variables on both sides