Problem 33 - Algebra for College Students

Question

-3 y=-x+3 $$

Step-by-Step Solution

Answer

The solution is \(y = \frac{x}{3} - 1\).

Step 1: Rewrite the Equation

The original equation is given as \[-3y = -x + 3\].Let's multiply every term by -1 to make the equation easier to handle:\[3y = x - 3\].

Step 2: Solve for y

To solve for \(y\), we need to isolate \(y\) by dividing all terms by 3:\[y = \frac{x}{3} - 1\].

Step 3: Verify the Solution

Let's substitute an arbitrary value for \(x\) to verify the rearranged equation.For example, if \(x = 3\), then:\[y = \frac{3}{3} - 1 = 1 - 1 = 0\].Check if this satisfies the original equation:\[-3(0) = -3 + 3 \Rightarrow 0 = 0\], which is true.

Key Concepts

Equation Solving

Linear equations form the foundation of understanding mathematics and are simple statements that show the equality of two expressions. Solving them can often be as straightforward as following a sequence of algebraic steps.
To get started, it's helpful to rearrange the equation to simplify it. In the case of the equation \(-3y = -x + 3\), notice that each term has a negative sign. This can be cumbersome, so multiplying through by \(-1\) gives us a cleaner and easier-to-manage equation: \(3y = x - 3\). This step doesn't change the equality; it simply transforms it into a more user-friendly format.
Helpful tips for equation solving:

• Always aim to simplify expressions on both sides of the equation.
• Remember that what you do to one side of the equation, you must do to the other to maintain balance.
• Focus on one side of the equation first, especially when simplifying complex terms.

Variable Isolation

Isolating the variable is a key step in equation solving. Your goal is to have the variable of interest, in this case, \(y\), alone on one side of the equation. After rewriting the equation, we have \(3y = x - 3\).
To isolate \(y\), divide all terms by 3: \(y = \frac{x}{3} - 1\). By performing this operation, we maintain the integrity of the equation while focusing on \(y\).
Reasons why variable isolation is important:

• It helps identify what the variable equals in terms of other variables or constants.
• Makes substitution of given values into an equation easier for verification or further analysis.
• Provides a clearer insight into how changes in other variables may affect \(y\).

Verification of Solutions

Once you've found a solution, it’s crucial to verify it to ensure its correctness. Let's revisit our solution: from \(y = \frac{x}{3} - 1\), we tested the solution with \(x = 3\).
Substitute \(x\) into the equation: \(y = \frac{3}{3} - 1 = 0\). We can now plug \(y = 0\) back into the original equation to see if it holds true: \(-3(0) = -3 + 3\) simplifies to \(0 = 0\). This confirms that our rearranged equation indeed satisfies the original given condition.
Key pointers for verifying solutions:

• Substitution is critical: try several values to test robustness.
• Verifying acts as a failsafe against algebraic errors that may have occurred during manipulation.
• Always conclude with comparing your final statement with typical mathematical identities like \(0 = 0\) to ensure accuracy.