Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation (if possible). $$\frac{3 x}{2}+\frac{1}{4}(x-2)=10$$

Step-by-Step Solution

Answer

The solution to the equation is \(x=6\).

Step 1: Remove Fractions

Multiplying every term by 4 to eliminate fractions: \(4*\frac{3x}{2}+4*\frac{1}{4}(x-2)=4*10\). This simplifies to: \(6x+x-2=40\).

Step 2: Simplify and Combine Like Terms

Combine the x terms on the left side of the equation: \(7x-2=40\).

Step 3: Isolate the Variable x

Add 2 to both sides of the equation: \(7x=42\).,

Step 4: Solve for x

Divide both sides by 7 to solve for x: \(x=6\).

Key Concepts

Elimination of Fractions

Fractions can complicate equations and make solving them less straightforward. To simplify the process, one effective technique is to eliminate the fractions. This is done by finding a common denominator and multiplying each term of the equation by this value. In the given exercise, the equation \( \frac{3x}{2} + \frac{1}{4}(x - 2) = 10 \) contains fractions with denominators of 2 and 4. The least common denominator here is 4.

By multiplying every term, including each fraction and whole number, by 4, you remove the fractions entirely. This transformation turns the original equation into \( 6x + x - 2 = 40 \). With the fractions out of the way, it becomes much easier to work through the equation and find the solution.

• Multiply each term by the least common denominator.
• Simplify to remove fractions completely.
• Proceed with easier terms in the equation.

Combining Like Terms

Once the fractions have been dealt with, the next step is simplifying the equation by combining like terms. This means bringing together terms that have the same variable. In our equation \( 6x + x - 2 = 40 \), the terms \( 6x \) and \( x \) are like terms because they both contain the variable \( x \).

By combining them, you sum up their coefficients to get \( 7x \), resulting in the equation \( 7x - 2 = 40 \). Simplifying equations by combining like terms helps in reducing the complexity to a simpler equation, making it easier to solve.

• Identify terms with the same variables.
• Combine coefficients of like terms.
• Simplify the equation further.

Isolation of Variable

The next step in solving an equation is to isolate the variable, which means getting the variable by itself on one side of the equation. This action grants you insight into what the variable actually equals. From the combined terms \( 7x - 2 = 40 \), start by getting rid of constant terms on the side containing the variable.

We accomplish this by adding 2 to both sides of the equation. It cancels out the \(-2\) on the left, giving us \( 7x = 42 \). This isolated variable with its coefficient can now be tackled easily.

• Move constants from the variable side by adding/subtracting.
• Ensure only the variable and its coefficient remain.
• Prep the equation for the final solving step.

Algebraic Manipulation

Finally, algebraic manipulation is used to solve for the variable by making it the subject of the equation. At the stage \( 7x = 42 \), the problem is to solve for \( x \). You'll usually do this by performing the inverse operation of whatever the coefficient of the variable does.

Here, the number 7 is multiplied by \( x \). To isolate \( x \), divide both sides by 7. This step yields \( x = 6 \). Once completed, you've solved for \( x \) using algebraic manipulation, which is the essential practice of altering equations to get answers.

• Identify the operation on the variable.
• Perform inverse operations to isolate the variable.
• Arrive at the solution of the equation.