Problem 47 - Introductory Algebra for College Students

Question

Solve each equation in using both the addition and multiplication properties of equality. Check proposed solutions. $$8 y+4=2 y-5$$

Step-by-Step Solution

Answer

The solution is \(y = - \frac{3}{2}\)

Step 1: Isolate the variable terms on one side

Our aim is to get all the terms with \(y\) on one side of the equation and the constants on the other. This can be achieved by using the addition/subtraction property of equality; if a term is subtracted or added on both sides of the equation, it doesn't change the equation. Start by subtracting \(2y\) from both sides: \[8y + 4 - 2y = 2y - 5 - 2y\]This simplifies to:\[6y + 4 = - 5\]

Step 2: Isolate the variable

Now, we subtract 4 from both sides to complete the isolation of \(y\):\[6y + 4 - 4 = -5 - 4\]Simplifying that gives us:\[6y = -9\]

Step 3: Solve for the variable

Finally, to solve for \(y\), we can use the multiplication/division property of equality. Dividing both sides of the equation by 6:\[\frac{6y}{6} = \frac{-9}{6}\]This gives us the solution:\[y = - \frac{3}{2}\]

Step 4: Check Solution

It's always a good idea to check our solution. Substitute \(y = - \frac{3}{2}\) back into the original equation:\[8(- \frac{3}{2}) + 4 = 2(- \frac{3}{2}) - 5\]Simplify to see if both sides are equivalent:\[-12 + 4 = -3 - 5\]\[-8 = -8\]The original equation holds true for \(y = - \frac{3}{2}\), therefore our solution is correct.

Key Concepts

Addition Property of Equality

When solving linear equations, one of the fundamental concepts we use is the addition property of equality. This property states that if you add or subtract the same value from both sides of an equation, the two sides remain equal. In our exercise, we started with the equation:\[8y + 4 = 2y - 5\]To begin isolating the variable, we used the addition property of equality by subtracting \(2y\) from both sides. The goal here was to have all the terms involving \(y\) on one side for easier manipulation:\[8y + 4 - 2y = 2y - 5 - 2y\]This simplified to:\[6y + 4 = -5\]By shifting terms around this way, we maintain the balance of the equation, making it steadily closer to revealing the solution for \(y\). This strategy is key when simplifying and handling equations.

Multiplication Property of Equality

After using the addition property to get all the variable terms on one side, the next logical step is solving for the variable. The multiplication property of equality allows us to do just that. This property states that multiplying or dividing both sides of an equation by the same non-zero number does not change the equality of the equation.In our equation:\[6y = -9\]We need \(y\) to stand alone. To achieve this, we divide both sides by \(6\):\[\frac{6y}{6} = \frac{-9}{6}\]Which simplifies to:\[y = -\frac{3}{2}\]By applying the multiplication property of equality, we've neatly isolated \(y\) and found its value. This powerful tool allows us to alter the form of the equation without altering its truth.

Checking Solutions

Checking your solution is a vital step in solving equations. This ensures that the value you've calculated as a solution truly satisfies the original equation. For our exercise, after calculating \(y = -\frac{3}{2}\), we substituted it back into the original equation:\[8(-\frac{3}{2}) + 4 = 2(-\frac{3}{2}) - 5\]Simplifying both sides gives:

• Left side: \[-12 + 4 = -8\]
• Right side: \[-3 - 5 = -8\]

Both sides equal \(-8\), confirming our calculated value for \(y\) is correct. Checking solutions is essential; it validates our solution process and boosts confidence in our final answer.

Isolation of Variables

The isolation of variables is a cornerstone of solving linear equations. To solve for a specific variable, you need to "isolate" it, or get it by itself on one side of the equation. We undertake this by systematically using properties of equality to move other terms away from the variable.In our problem, we isolated \(y\) by removing other terms step by step:

• Subtracted \(2y\) from both sides: \(6y + 4 = -5\)
• Subtracted \(4\) from both sides: \(6y = -9\)
• Divided both sides by \(6\): \(y = -\frac{3}{2}\)

Each of these steps brought us closer to having just \(y\) alone on one side of the equation. Isolation involves strategic manipulation, crucial for unpacking and understanding the structure of the equation.