Problem 16 - Prealgebra

Question

Solve each equation. Check your solution. $$4(x-2)=3(1.5+x)$$

Step-by-Step Solution

Answer

The solution is \(x = 12.5\) and it checks out.

Step 1: Distribute the Constants

Apply the distributive property to each side of the equation to remove the parentheses. The left side becomes \(4x - 8\) and the right side becomes \(4.5 + 3x\). The equation is now \(4x - 8 = 4.5 + 3x\).

Step 2: Isolate Variable Terms

Move all terms containing the variable to one side of the equation and constant terms to the other side. Subtract \(3x\) from both sides to get \(4x - 3x - 8 = 4.5\). This simplifies to \(x - 8 = 4.5\).

Step 3: Solve for the Variable

Add 8 to both sides of the equation to isolate \(x\). This gives \(x = 4.5 + 8\), which simplifies to \(x = 12.5\).

Step 4: Check the Solution

Substitute \(x = 12.5\) back into the original equation to verify the solution. The original equation is \(4(x-2) = 3(1.5+x)\). Substitute to get \(4(12.5-2) = 3(1.5+12.5)\), which simplifies to \(4(10.5) = 3(14)\). Both sides equal 42, confirming that the solution is correct.

Key Concepts

Distributive Property

The distributive property is a fundamental principle used in algebra. It's like sharing out a number over terms inside parentheses. So, if you have an expression like \(a(b + c)\), you can distribute the \(a\) over both \(b\) and \(c\). This means you rewrite it as \(ab + ac\). Let's apply this in our equation:

• Original equation: \(4(x-2) = 3(1.5+x)\)
• Distribute 4 to both terms inside \((x-2)\) to get \(4x - 8\).
• Distribute 3 to both terms inside \((1.5+x)\) to get \(4.5 + 3x\).

The equation now looks like \(4x - 8 = 4.5 + 3x\). This step simplifies the expressions and removes the parentheses, making it easier to work with the equation.

Isolating Variables

Isolating the variable means getting the variable all by itself on one side of the equation. It is a crucial step to solve an equation. In our exercise, to isolate \(x\), we need to separate terms with \(x\) from the numbers:

• We start with \(4x - 8 = 4.5 + 3x\).
• Subtract \(3x\) from both sides: \(4x - 3x - 8 = 4.5\).
• This becomes \(x - 8 = 4.5\).

Now, \(x\) is getting closer to being by itself. It's important to handle each step carefully to maintain balance in the equation, just like a perfectly balanced scale.

Solving Linear Equations

Once you've isolated the variable term, the next step is to solve for the variable completely. With \(x\) almost by itself in \(x - 8 = 4.5\), you'll want to eliminate the constant next to it:

• Add 8 to both sides to move the 8: \(x - 8 + 8 = 4.5 + 8\).
• Simplifying this gives \(x = 12.5\).

We've found that \(x\) is equal to 12.5. Solving linear equations often means performing these small arithmetic operations to isolate \(x\) completely. Once you do, you've found your solution!

Checking Solutions

Once you think you've found the solution, it's essential to check your work. Plugging the solution back into the original equation ensures it truly satisfies it:

• Original equation: \(4(x-2) = 3(1.5+x)\).
• Substitute \(x = 12.5\) into the equation: \(4(12.5-2) = 3(1.5+12.5)\).
• Simplify both sides: \(4 \times 10.5 = 3 \times 14\).
• Both sides equal 42, confirming the accuracy.

If both sides are equal after substitution, it validates your solution. This process reassures that you've done everything correctly and didn't make any errors along the way.