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Magazine Chapter 6: Problem 2
Solve each equation and check the solution. $$\frac{2 y}{5}-8=\frac{4 y}{5}$$
Short Answer
Expert verified
The solution to the equation \(\frac{2y}{5}-8=\frac{4y}{5}\) is \(y = -20\).
Step by step solution
01
Simplify Equation
Start by rearranging terms on both sides of the equation: subtract \(\frac{4y}{5}\) from both sides to isolate \(y\) terms. \(\frac{2y}{5} - \frac{4y}{5} = -8\). The equation simplifies to: \(-\frac{2y}{5} = -8\).
02
Solve for y
To solve for \(y\), multiply both sides of the equation by -5: \[-\frac{2y}{5} * -5 = -8 * -5\]. The equation simplifies to: \(2y = 40\).
03
Final Solution
Finally, divide both sides by 2: \(\frac{2y}{2} = \frac{40}{2}\). This simplifies to: \(y = 20\).
04
Check
Check by substituting \(y = 20\) into the equation: \(\frac{2*20}{5}-8=\frac{4*20}{5}\). Simplify the expression: \(8=16\). The equation is not true, thus, there's a mistake at step 2 which needs to be corrected.
05
Correct Mistake and Solve for y
In Step 2, multiplying by -5 should be performed to only \(-\frac{2y}{5}\) term which results in \(2y = 40\). Not to the -8. So the correct equation from step 2 should be: \(2y = -40\). Now if we proceed as before, dividing both sides by 2 results in : \(y = -20\).
06
Correct Check
Now, check by substituting \(y = -20\) into the equation: \(\frac{2*(-20)}{5}-8=\frac{4*(-20)}{5}\). Simplify the expression: \(-8=-16\). The equation is true, so \(y = -20\) is the correct solution.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Solving Equations
To solve an algebraic equation like \( \frac{2y}{5} - 8 = \frac{4y}{5} \), the core goal is to find the value of \( y \) that makes the equation true. This involves a series of systematic steps.
- Rearranging Terms: Identify like terms on each side of the equation and move them to one side. For the equation given, you start by moving all terms involving \( y \) to one side, which is achieved by subtracting \( \frac{4y}{5} \) from both sides.
- Isolating the Variable: The equation simplifies to \( -\frac{2y}{5} = -8 \). Here, the objective is to have \( y \) alone on one side.
- Solving for \( y \): You will then perform operations to isolate \( y \, \) like multiplying both sides of the equation to remove fractions and solve for the variable.
Checking Solutions
After solving an equation and arriving at a solution, it's important to verify that the solution is correct. This step is often overlooked but is vital to ensure accuracy.
- Substituting Back: Substitute the value back into the original equation. For instance, if you found \( y = -20 \), check this by substituting \( y \) into both sides of \( \frac{2y}{5} - 8 = \frac{4y}{5} \).
- Verify Equality: Simplify the expressions on both sides to confirm that the left-hand side equals the right-hand side. For every correct solution, substituting should make both sides of the equation equal.
The Simplification Process
Simplification is a fundamental aspect of solving algebraic equations. It involves reducing complex expressions to simpler forms, making calculations easier.
- Combine Like Terms: In our case, after moving the terms involving \( y \), combine them. As seen, this meant finding the difference \( \frac{2y}{5} - \frac{4y}{5} = -\frac{2y}{5} \).
- Eliminate Fractions: Once like terms are combined, the next step is to eliminate fractions for easier computation. Multiply through by the denominator, as seen when \( -\frac{2y}{5} = -8 \) became \( 2y = 40 \) after multiplying by 5.
Integer Division
Integer division is particularly useful when you're solving equations involving fractions. It involves dividing two integers to find a quotient.
- Conceptual Clarity: Integer division results in an integer value, discarding any remainder. In the problem, once \( 2y = 40 \) was reached, dividing both sides by 2 gave \( y = 20 \).
- Correct Application: Missteps in division can occur if one rushes or handles negatives incorrectly. As seen, correcting from \( 2y = 40 \) to \( 2y = -40 \) and then \( y = -20 \) showed how essential precise division is to find the correct solution.
