Problem 57 - Algebra 2

Question

Solve each equation. Check your solution. \(\frac{5}{7} y-3=\frac{3}{7} y+1\)

Step-by-Step Solution

Answer

The solution is \(y = 14\).

Step 1: Eliminate Fractions

To eliminate the fractions, multiply every term in the equation by 7, which is the common denominator of all terms. This gives: \[ 7 \times \left( \frac{5}{7} y - 3 \right) = 7 \times \left( \frac{3}{7} y + 1 \right) \] Simplifying each side: \[ 5y - 21 = 3y + 7 \]

Step 2: Collect Like Terms

Move all terms involving \(y\) to one side and constant terms to the other. Subtract \(3y\) from both sides: \[ 5y - 3y - 21 = 7 \] Which simplifies to: \[ 2y - 21 = 7 \]

Step 3: Solve for y

Add 21 to both sides to isolate the term with \(y\): \[ 2y = 7 + 21 \] Simplifying gives: \[ 2y = 28 \] Divide both sides by 2 to solve for \(y\): \[ y = \frac{28}{2} \] \[ y = 14 \]

Step 4: Check the Solution

Substitute \(y = 14\) back into the original equation to verify: \[ \frac{5}{7} \times 14 - 3 = \frac{3}{7} \times 14 + 1 \] Calculate each side: \[ 10 - 3 = 6 + 1 \] Simplifying both sides: \[ 7 = 7 \] Both sides are equal, verifying that \(y = 14\) is the correct solution.

Key Concepts

Fractions

Fractions can make solving linear equations a bit tricky at first glance. A fraction consists of a numerator and a denominator, and when you encounter fractions in an equation, it adds an extra layer of complexity. In the given equation \(\frac{5}{7} y - 3 = \frac{3}{7} y + 1\), fractions are present on both sides. Understanding how to handle these fractions is key. You can think of fractions as divisions—\(\frac{5}{7} y\) is the same as \(y\) divided by 7, multiplied by 5. To solve equations like these, it is often helpful to eliminate fractions right at the start, which simplifies the equation greatly.

Common Denominator

A common denominator is a shared multiple of the denominators of the fractions involved in an equation. In our example, both fractions have a denominator of 7. Hence, the common denominator is clearly 7. But why is this important?Finding a common denominator allows us to eliminate the fractions by multiplying each term by this number. This greatly simplifies the equation, converting it into a form without fractions. In our original equation, multiplying through by 7 transforms it from fractional form to a simpler linear equation: - Multiplying every term by 7 gives \[ 7 \times \left( \frac{5}{7} y - 3 \right) = 7 \times \left( \frac{3}{7} y + 1 \right) \] - Resulting in: \(5y - 21 = 3y + 7\).Now, the equation is much easier to handle.

Isolate Variable

The goal of solving any equation is to find the value of the unknown variable—in this case, \(y\). Isolating the variable means getting \(y\) by itself on one side of the equation.After removing the fractions, we end up with \(5y - 21 = 3y + 7\). The next steps involve a series of operations to get \(y\) alone. Here's how you do it:- Move all \(y\)-terms to one side: subtract \(3y\) from both sides, resulting in \(2y - 21 = 7\).- Then, move the constants to the opposite side. Add 21 to both sides to simplify further: \(2y = 28\).- Finally, divide by the coefficient of \(y\) which is 2: - We divide both sides by 2: \(y = \frac{28}{2} \), simplifying to \(y = 14\).Now \(y\) is isolated, and you've found its value.

Check Solution

Once you've solved for \(y\), it's important to verify the solution by substituting it back into the original equation. This step ensures there were no mistakes during the calculation process.For our problem, we have \(y = 14\). We substitute it back into the original equation: - \(\frac{5}{7} \times 14 - 3 = \frac{3}{7} \times 14 + 1\).- Simplify both sides: - On the left, \(10 - 3\) equals 7. - On the right, \(6 + 1\) also equals 7.Both sides of the equation are equal, confirming that \(y = 14\) is indeed the correct solution. Always make sure this final check is done; it's a vital part of solving equations accurately!