Problem 34 - Algebra 2

Question

Solve each equation. Check your solution. \(x+9 x-6 x+4 x=20\)

Step-by-Step Solution

Answer

The solution is \(x = 2.5\).

Step 1: Simplify the Equation

Combine like terms on the left side of the equation. You have the terms \(x\), \(9x\), \(-6x\), and \(4x\) which are all multiples of \(x\). Simplifying gives:\[x + 9x - 6x + 4x = (1 + 9 - 6 + 4)x = 8x\]So, the equation becomes \(8x = 20\).

Step 2: Solve for x

Now, solve for \(x\) by dividing both sides of the equation by 8:\[8x = 20\]\[x = \frac{20}{8}\]Simplify the fraction:\[x = \frac{5}{2}\] or \(x = 2.5\).

Step 3: Check the Solution

Substitute \(x = 2.5\) back into the original equation to verify the solution:\(x + 9x - 6x + 4x = 20\)Replace \(x\) with 2.5:\(2.5 + 9(2.5) - 6(2.5) + 4(2.5)\)Calculate each term:2.5 + 22.5 - 15 + 10 = 20Since the left side equals the right side, \(x = 2.5\) is correct.

Key Concepts

Combining Like Terms

Combining like terms is an essential concept when solving linear equations. Like terms are terms in an equation that have the same variable raised to the same power. For example, in the expression \(x + 9x - 6x + 4x\), all terms are like terms because they involve the variable \(x\). The key to combining these terms is to sum up their coefficients. So, you add \(1\), \(9\), \(-6\), and \(4\) together, which results in \(8x\). This step simplifies the problem and reduces the original expression to a more manageable form. Simplification makes it easier to solve the equation, as you'll be dealing with fewer terms.

• Identify all like terms.
• Add or subtract their coefficients.
• Rewrite the expression using the simplified coefficients.

Simplifying Fractions

Once you have combined like terms, the next step often involves solving for the variable, which may result in a fraction. Simplifying fractions is crucial in writing the simplest form of your solution. In our example, we solved for \(x\) and obtained \(x = \frac{20}{8}\). Both 20 and 8 share a common factor, 4. By dividing the numerator and the denominator by this common factor, you simplify the fraction to \(x = \frac{5}{2}\). This simplification shows the ratio in its simplest form.

• Identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and denominator by the GCD.
• Rewrite the fraction in its simplest form.

Verification of Solutions

Verifying your solutions is a valuable step to ensure that your answer is correct. After finding \(x = 2.5\), substitute it back into the original equation to check your work: \(x + 9x - 6x + 4x = 20\). Replace every \(x\) with 2.5: \(2.5 + 9(2.5) - 6(2.5) + 4(2.5) = 20\). Now calculate: \(2.5 + 22.5 - 15 + 10\). The results sum up to 20, confirming that \(x = 2.5\) is indeed correct. Verification helps to reduce mistakes and confirm the solution is consistent with the original equation.

• Substitute your solution back into the original equation.
• Calculate each term to verify.
• Ensure both sides of the equation are equal.