Problem 49 - Elementary and Intermediate Algebra

Question

Use a check to determine whether \(\frac{21}{5}\) is a solution of: \(x+20=4 x-1+2 x\)

Step-by-Step Solution

Answer

Yes, \(\frac{21}{5}\) is a solution because both sides of the equation equal \(\frac{121}{5}\).

Step 1: Identify the Equation

The given equation is \(x + 20 = 4x - 1 + 2x\). We need to determine if \(\frac{21}{5}\) is a solution to this equation.

Step 2: Simplify the Right Side

Combine like terms on the right side of the equation. \(4x - 1 + 2x\) simplifies to \(6x - 1\). Thus, the equation becomes \(x + 20 = 6x - 1\).

Step 3: Substitute the Value

Substitute \(x = \frac{21}{5}\) into the equation to determine if both sides are equal.- The left side becomes: \(\frac{21}{5} + 20\).- The right side becomes: \(6 \left(\frac{21}{5}\right) - 1\).

Step 4: Calculate the Left Side

Calculate \(\frac{21}{5} + 20\).- Convert 20 to a fraction: \(20 = \frac{100}{5}\).- Add the fractions: \(\frac{21}{5} + \frac{100}{5} = \frac{121}{5}\).

Step 5: Calculate the Right Side

Calculate \(6 \left(\frac{21}{5}\right) - 1\).- Multiply: \(6 \times \frac{21}{5} = \frac{126}{5}\).- Convert \(-1\) to a fraction: \(-1 = -\frac{5}{5}\).- Subtract: \(\frac{126}{5} - \frac{5}{5} = \frac{121}{5}\).

Step 6: Compare Both Sides

Both the left and right sides are equal to \(\frac{121}{5}\), confirming that \(x = \frac{21}{5}\) makes both sides of the equation equal.

Key Concepts

Checking Solutions

When solving an equation, checking the solution is a crucial step that ensures accuracy. This process confirms whether a proposed solution truly satisfies the equation. Let's break it down step-by-step:

• Identify the Equation: Start by clearly outlining the equation you are working with. In our example, the equation is \(x + 20 = 4x - 1 + 2x\).
• Substitute Your Value: Insert the value you are checking into the equation. Here, we use \(x = \frac{21}{5}\).
• Evaluate Both Sides: Calculate the expressions on both sides of the equation separately after substitution. Each needs to yield the same value for the solution to be valid.
• Compare the Results: If both sides of the equation are equal, the proposed value is a solution. If they are not, it isn't a solution.

Ultimately, checking the solution reassures you of the accuracy of a problem's outcome, preventing errors before proceeding further.

Combining Like Terms

Combining like terms is a fundamental technique in algebra that simplifies equations, making them easier to solve. Here's how it works:Like terms are terms that have the same variable raised to the same power. For instance, \(4x\) and \(2x\) are like terms, whereas \(4x\) and \(4y\) are not.

• Identify Like Terms: Look across the entire equation to find terms that have the same variable component. In our exercise, \(4x\) and \(2x\) on the right side of the equation are like terms.
• Add or Subtract Coefficients: Once identified, add or subtract the coefficients of these like terms. Combining \(4x + 2x\) gives us \(6x\).
• Simplify the Equation: By combining like terms, the equation becomes more straightforward and manageable. This step reduces complexity and allows us to focus on solving for the unknown variable with fewer terms in play.

Simplifying through combining like terms is essential, especially in equations with multiple variables, as it reduces potential confusion and errors.

Fractions in Algebra

Fractions can often pose a challenge in algebra, but they are simply another way to express division. Proper handling of fractions is vital for solving equations. Here's a guide on dealing with fractions:

• Fractions as Division: Remember that a fraction represents division. For instance, \(\frac{21}{5}\) indicates 21 divided by 5.
• Converting Whole Numbers to Fractions: When working with equations, whole numbers may need converting into fractions to perform arithmetic easily. For example, converting 20 to \(\frac{100}{5}\) ensures we can add it to \(\frac{21}{5}\).
• Adding and Subtracting Fractions: Combine fractions by ensuring they have a common denominator. In our example, both fractions are over 5, allowing for straightforward addition or subtraction.
• Multiplication and Division: Multiplying fractions involves multiplying across the numerators and denominators directly. Simplify when possible to keep calculations manageable.

Fractions in algebra require careful attention, but understanding their properties can greatly enhance your ability to solve complex equations effectively. Mastery of fractions paves the way for solving both simple and advanced algebraic problems.