Problem 46 - Elementary Algebra

Question

Solve. $$ 22 x-55=-22 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{3}{2} \).

Step 1: Isolate the Variable

To solve for \( x \), we need to isolate it on one side of the equation. Start by adding 55 to both sides of the equation to move the constant term from the left side.\[22x - 55 + 55 = -22 + 55 \22x = 33\]

Step 2: Solve for x

Now, divide both sides of the equation by 22 to solve for \( x \).\[x = \frac{33}{22}\]

Step 3: Simplify the Fraction

Simplify \( \frac{33}{22} \) by finding the greatest common divisor for the numerator and the denominator. The greatest common divisor of 33 and 22 is 11, so divide both by 11:\[x = \frac{33 \div 11}{22 \div 11} = \frac{3}{2}\]

Step 4: Conclusion: Verification

To ensure the solution is correct, substitute \( x = \frac{3}{2} \) back into the original equation:\[22\left(\frac{3}{2}\right) - 55 = 11 - 55 = -22\]Since the original equation is satisfied, our solution is verified.

Key Concepts

Isolate the Variable

When solving linear equations like \(22x - 55 = -22\), the goal is to find the value of \(x\) that makes the equation true. This requires isolating the variable \(x\) on one side of the equation. Think of it like cleaning up a messy room until all distractions are gone.Start by getting rid of the constant \(-55\) that is hanging out with \(x\) on the left side. This is done by performing the opposite operation: adding \(55\) to both sides. That gives us:

• Left Side: \(22x - 55 + 55 = 22x\)
• Right Side: \(-22 + 55 = 33\)

Now we have an equation \(22x = 33\), where the variable is more isolated. From here, you can see what \(x\) should be without any number distracting it.

Simplify Fractions

After isolating the variable, you might end up with a fraction. Remember, fractions are just divisions in disguise! In our case, we found \(x = \frac{33}{22}\).A key step in math is simplifying fractions. It involves making the numbers as small as possible, while still keeping the same value. This simplifies the work ahead and makes understanding easier. The trick here is finding the greatest common divisor (GCD).For 33 and 22, the GCD is 11. Simplifying \(\frac{33}{22}\) means:

• Divide 33 by 11: \(33 \div 11 = 3\)
• Divide 22 by 11: \(22 \div 11 = 2\)

So \(\frac{33}{22}\) simplifies to \(\frac{3}{2}\). A simpler fraction makes your solution neater and more elegant!

Verification of Solution

Verifying your solution ensures there's no mistake in the math. It’s like double-checking your work before you submit it.Once we have \(x = \frac{3}{2}\), substitute this value back into the original equation to check if the left side equals the right side:Substitute and calculate:

• Calculate \(22 \times \frac{3}{2} = 33\).
• Subtract the 55: \(33 - 55 = -22\).

This matches the right side of the equation \(-22\), confirming our solution is correct. Verification gives you confidence that you've done everything properly, ensuring there’s no error in your calculations.