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.
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Now Playing: Solve linear equations using distributive property – Example 0aJohn has a round table with a circumference of 314.16 cm, but it is too big for his new home. So, he cut off a 10 cm wide border around the edge.
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Distributive property is an algebra property that we use all the time! When you see equations in the form of a(x+b), you can transform them into ax+ab by multiplying the terms inside a set of parentheses. In this section, we will make use of this property to help us solve linear equations.
Welcome to our lesson on solving two-step linear equations using the distributive property! This fundamental skill is crucial for advancing your algebra knowledge. To kick things off, we've prepared an introduction to two-step linear equations video that will give you a clear visual understanding of the concept. This video is an essential starting point, as it breaks down the process step-by-step, making it easier to grasp. As we dive into two-step linear equations, you'll see how the distributive property plays a key role in simplifying expressions. We'll learn to "unpack" terms within parentheses, which is often the first step in solving these equations. Don't worry if it seems challenging at first with practice, you'll become more comfortable applying this property. Remember, mastering this technique will set a strong foundation for more complex algebraic problems in the future. Let's get started on this exciting journey through two-step linear equations!
Solve the equation using model. 4 ( x + 1 ) = 12 4\left( \right) = 12 4 ( x + 1 ) = 12
To solve the equation 4 ( x + 1 ) = 12 4(x + 1) = 12 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 x + 1 x + 1 and that these groups together equal 12. Our goal is to find the value of x x x .
We start by representing the left side of the equation visually. We have four groups of x + 1 x + 1 x + 1 . This means we need to draw four sets of x x x tiles and four sets of +1 tiles. Each group looks like this: x + 1 x + 1 x + 1 .
So, we draw four groups:
On the right side of the equation, we have 12 positive units. We represent this by drawing 12 positive tiles.
Next, we simplify the visual representation. We combine all the x x x tiles and all the +1 tiles on the left side. This gives us:
So, the left side of the equation is now represented as 4 x + 4 4x + 4 4 x + 4 . The right side remains as 12 positive tiles.
To isolate the variable x x 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 4 x 4x 4 x . On the right side, we subtract 4 from 12, which gives us 8 positive tiles.
So, the equation now looks like this: 4 x = 8 4x = 8 4 x = 8 .
Now, we need to find the value of x x x . We have 4 x x x tiles that equal 8 positive tiles. To find the value of one x x x tile, we divide the 8 positive tiles into 4 equal groups.
Each group will have 2 positive tiles. Therefore, each x x x tile is worth 2 positive tiles.
So, x = 2 x = 2 x = 2 .
By using a visual model, we have successfully solved the equation 4 ( x + 1 ) = 12 4(x + 1) = 12 4 ( x + 1 ) = 12 . The value of x x 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:
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.
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.
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.
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.
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.
PrerequisitesUnderstanding 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.
