Problem 28 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation (if possible). $$\frac{3 x}{4}+\frac{x}{2}=-5$$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{3 x}{4}+\frac{x}{2}=-5\) is \(x = -4\).

Step 1: Combine Like Terms

Simplify the left side of the equation by combining the terms with x. Both \(\frac{3 x}{4}\) and \(\frac{x}{2}\) contain 'x'. A term with an 'x' can be combined with other terms that contain 'x' as they are like terms. To do this, find a common denominator and add the fractions to simplify: \(\frac{3 x}{4}+\frac{x}{2} = \frac{3x + 2x}{4} = \frac{5x}{4}\). So, the simplified equation becomes: \(\frac{5x}{4} = -5\)

Step 2: Isolate x

Now, isolate x by getting rid of the fraction. The fraction can be removed by multiplying both sides of the equation by 4: \(4 * \frac{5x}{4} = -5 * 4\). This simplifies to: \(5x = -20\)

Step 3: Solve for x

Finish solving the equation by getting x alone. This is done by dividing both sides of the equation by 5: \( \frac{5x}{5} = \frac{-20}{5}\). This yields the following: \(x = -4\), which is the solution to the equation.

Key Concepts

Combining Like Terms

When solving linear equations, combining like terms is an essential technique. This means putting together terms that have the same variables raised to the same power. In the given equation, \( \frac{3x}{4} + \frac{x}{2} = -5 \), both terms on the left side contain the variable \( x \). They are like terms because they share the same variable, which allows us to combine them into a single expression. This simplification is crucial because it reduces the equation to a more manageable form, making it easier to solve.

Common Denominator

Working with fractions in equations often requires a common denominator. A common denominator allows you to combine fractions by making the denominators the same. In the example, fractions \( \frac{3x}{4} \) and \( \frac{x}{2} \) need attention.

• Identify the least common denominator: In this case, it's 4 since 4 is a multiple of 2.
• Convert each fraction: Multiply the numerator and denominator of \( \frac{x}{2} \) by 2 to achieve \( \frac{2x}{4} \).

Now that the fractions are \( \frac{3x}{4} \) and \( \frac{2x}{4} \), you can easily combine them as \( \frac{5x}{4} \). This step is a powerful tool in managing and simplifying equations that contain fractions.

Isolating the Variable

Isolating the variable involves rearranging the equation so that the variable is by itself on one side. In our simplified equation \( \frac{5x}{4} = -5 \), the goal is to isolate \( x \). This requires eliminating the fraction by performing inverse operations. Here, we eliminate the denominator by multiplying both sides by 4. This gives us \( 5x = -20 \).

• Why multiply by 4? Because it’s the inverse operation of dividing by 4, which simplifies the equation.
• Achieving \( 5x = -20 \) is a significant step as it linearizes the equation further.

Isolating the variable is crucial for finding the solution as it sets the stage to easily solve for the unknown.

Solving Equations

The final step in solving equations focuses on simplifying to a point where you can determine the value of the variable. Once the equation is streamlined from \( 5x = -20 \), the aim is to get \( x \) on its own. To accomplish this, divide both sides by 5, the coefficient of \( x \):

• Divide \( 5x \) by 5 to get \( x \).
• Divide \( -20 \) by 5 to find the numerical value of \( x \).

This gives \( x = -4 \). Solving for \( x \) tells us the value of \( x \) that satisfies the original equation. This process of solving linear equations is foundational in mathematics, as it not only identifies solutions but also builds understanding of the relationships between values and operations within the equation.