Problem 108
Question
Solve equation. \(2(3 x+4)=3 x+2[3(x-1)+2]\)
Step-by-Step Solution
Verified Answer
The solution to the equation is \(x=\frac{10}{3}\)
1Step 1: Distribute the multiplication
First, distribute the multiplication on both sides of the equation to simplify it: \(2(3 x+4)=6x+8\). And \(3 x+2[3(x-1)+2] = 3x+2(3x-3+2)=3x+2(3x-1)=3x+6x-2=9x-2\). The equation now reads as \(6x+8=9x-2\)
2Step 2: Combine like terms
Next, combine like terms on each side of the equation. The equation remains the same \(6x+8=9x-2\). Now, subtract \(6x\) from both sides: \(6x - 6x + 8 = 9x - 6x - 2\), which simplifies to: \(8 = 3x - 2\).
3Step 3: Solve for X
Add 2 to both sides of the equation to isolate \(x\). The equation becomes: \(8 + 2 = 3x - 2 + 2\), so \(10 = 3x\). Finally, divide by 3 on both sides, resulting \(x=\frac{10}{3}\)
Key Concepts
Understanding the Distributive PropertyCombining Like TermsIsolating the Variable
Understanding the Distributive Property
When solving linear equations, the distributive property plays a vital role. It's a tool that lets you multiply a single term by each term inside a set of parentheses. In our example, we used the distributive property to expand the expressions on both sides of the equation.
This means instead of solving for something like \(2(3x+4)\), you apply the property: \(2 \times 3x\) plus \(2 \times 4\), which simplifies to \(6x+8\). Similarly, \(2[3(x-1)+2]\) becomes \(2(3x-3+2)\), which then simplifies to \(6x-2\) when we distribute the 2. It's essential to perform this step correctly, as it sets the stage for combining like terms and ultimately solving for the variable.
This means instead of solving for something like \(2(3x+4)\), you apply the property: \(2 \times 3x\) plus \(2 \times 4\), which simplifies to \(6x+8\). Similarly, \(2[3(x-1)+2]\) becomes \(2(3x-3+2)\), which then simplifies to \(6x-2\) when we distribute the 2. It's essential to perform this step correctly, as it sets the stage for combining like terms and ultimately solving for the variable.
Combining Like Terms
Once the distributive property has been applied, the next step involves combining like terms. These are terms that contain the same variable raised to the same power. In our worked solution, after distributing, we have the equation \(6x+8=9x-2\).
Like terms on different sides of the equal sign can be balanced by performing the same operation on both sides. Subtracting \(6x\) from each side gives us \(8=3x-2\). This step simplifies the equation and moves us closer to isolating the variable, making the equation easier to solve. Remember, maintaining balance is crucial: what you do to one side of the equation, you must do to the other.
Like terms on different sides of the equal sign can be balanced by performing the same operation on both sides. Subtracting \(6x\) from each side gives us \(8=3x-2\). This step simplifies the equation and moves us closer to isolating the variable, making the equation easier to solve. Remember, maintaining balance is crucial: what you do to one side of the equation, you must do to the other.
Isolating the Variable
The final step in solving a linear equation is to isolate the variable. This means getting the variable by itself on one side of the equation with a coefficient of 1. After combining like terms, our equation looked like this: \(8 = 3x-2\).
Now, we want to get \(x\) alone. First, add 2 to both sides to cancel out the -2, resulting in \(10 = 3x\). Lastly, divide both sides by 3 to find \(x\) by itself. The solution is \(x = \frac{10}{3}\). Isolating the variable is the goal of solving equations; it provides the answer to the problem in its simplest form.
Now, we want to get \(x\) alone. First, add 2 to both sides to cancel out the -2, resulting in \(10 = 3x\). Lastly, divide both sides by 3 to find \(x\) by itself. The solution is \(x = \frac{10}{3}\). Isolating the variable is the goal of solving equations; it provides the answer to the problem in its simplest form.
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