Problem 70 - Elementary Algebra

Question

Solve the equation \(\frac{4 x}{5}=-3\).

Step-by-Step Solution

Answer

Answer: The value of x in the given equation is $x = \frac{-15}{4}$.

Step 1: Multiply both sides by the reciprocal of the fraction's denominator

To eliminate the fraction, we multiply both sides of the equation by the reciprocal of the denominator, which in this case is \(5\): \((5) \cdot \left(\frac{4x}{5}\right) = -3 \cdot (5)\).

Step 2: Simplify

We can now simplify both sides of the equation. Multiplying both sides by 5, we get: \(4x = -15\)

Step 3: Solve for x

At this point, we need to isolate the variable \(x\). To do this, we divide both sides of the equation by 4: \(x = \frac{-15}{4}\).

Step 4: Final Answer

The solution to the given equation is \(x=\frac{-15}{4}\).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that allows us to find the values of variables that make an equation true. An equation is like a balance scale, where both sides should be equal to each other. The goal is to keep this balance while working to isolate the variable. To solve an equation successfully, you typically perform operations that simplify the equation step-by-step. These operations include:

• Addition and subtraction to move terms from one side to the other.
• Multiplication and division to eliminate fractions or coefficients of the variable.

A helpful tip is to do the inverse operation to both sides of the equation to maintain its balance. By carefully applying these principles, you can solve even the most complex algebraic equations.

Fractions in Equations

Fractions often appear in algebraic equations and can make solving them seem challenging. However, handling fractions in equations is much easier if you remember a few key steps.
To remove a fraction from an equation, you can multiply both sides by the denominator of the fraction. This step eliminates the fraction, transforming the equation into a simpler form. For example, in the equation \(\frac{4x}{5} = -3\), multiplying both sides by \(5\) turns it into \(4x = -15\).
Always ensure you are applying the multiplication correctly to each term in the equation to avoid errors. Once the fraction is removed, you can proceed with solving the equation using simpler arithmetic. Fractions not only test your understanding of basic operations but also instill a deeper grasp of algebraic manipulation.

Multiplication in Equations

Multiplication plays a crucial role in solving equations, especially when dealing with fractions. When an equation involves fractions, multiplying both sides by the denominator simplifies the problem significantly. This technique is known as "clearing the fraction."
Take the equation \(\frac{4x}{5} = -3\) as an example. By multiplying both sides of the equation by \(5\), you eliminate the fraction:

• The left side becomes \(4x\), simplifying your work.
• The right side becomes \(-15\), making it easier to isolate \(x\).

After clearing the fraction, you might need to multiply or divide further to solve for the variable. The rule of thumb is to apply arithmetic operations that move you closer to isolating the variable. Using multiplication appropriately ensures your equation remains balanced and leads you directly to the solution. This understanding builds a strong foundation for tackling more advanced algebra problems.