Problem 32 - Algebra 1

Question

Evaluate the expr$$ \frac{3}{5} x-y \text { when } x=-25 \text { and } y=-10 $$ssion.

Step-by-Step Solution

Answer

-5

Step 1: Substitution of the given values

Substitute \(x = -25\) and \(y = -10\) into the expression \(\frac{3}{5}x - y\). So, the expression becomes \(\frac{3}{5}*(-25) - (-10)\).

Step 2: Simplifying the multiplication

Next, simplify the expression by carrying out the multiplication. This gives us: \(-15 - (-10)\).

Step 3: Final simplification

Simplify the expression by carrying out the subtraction operation. The two minus signs become a plus, thus the final simplified expression will be: \(-15 + 10\).

Step 4: Evaluating the expression

Finally, evaluating the simplified expression, we have: \(-15 + 10 = -5\) as the value of the expression.

Key Concepts

Substitution

The first step in evaluating an expression is substitution, which involves plugging in the given values for the variables. For example, in our original expression, \[ \frac{3}{5}x - y, \] the variables are \(x\) and \(y\). We were told that \(x = -25\) and \(y = -10\). We substitute these into the expression, changing it to \[ \frac{3}{5}*(-25) - (-10). \] This step sets the stage for simplification by ensuring that each part of the expression has a specific numerical value.

• Identify variables in your expression.
• Substitute each variable with its given value.
• Rewrite the expression with the numbers in place of variables.

Substitution is essential because it transforms an abstract expression into one that you can compute.

Multiplication

Once you have substituted the values, the next step involves multiplication. Multiplication is a key element in evaluating expressions, especially when fractions are present.In this instance, after substitution, we find ourselves with the expression \[ \frac{3}{5} * (-25). \] Multiplying these values requires you to multiply the numerator by the value and then divide by the denominator, resulting in: \[ -15. \]

• Multiply the numerator of the fraction with the value.
• Divide the result by the fraction's denominator.

For negative numbers, multiplying by a negative value flips the sign. This step reduces complexity by working on one part of the expression at a time.

Simplification

The simplification stage is where you reduce the expression further, putting it into its simplest form.After completing the multiplication step, the expression is \[ -15 - (-10). \] The goal here is to perform operations involving subtraction and addition to get to a basic numerical form. In this case, simplifying involves handling the double negatives: \[ -15 - (-10) = -15 + 10. \]

• Combine similar terms if possible.
• Resolve double negatives by converting subtraction into addition.
• Use basic arithmetic operations.

Simplification helps in making complex expressions understandable and gives a clearer path to finding the final value.

Negatives

Negative numbers can sometimes be tricky, especially in expressions involving double negatives or subtraction.Our example illustrates how these can appear: \[ -15 - (-10). \] Notice the two negatives—one with the 15 and another in front of the 10.When you subtract a negative, it's equivalent to adding the positive of that number. These two negatives effectively cancel out: \[ -15 - (-10) = -15 + 10. \]

• Double negatives become addition.
• Recognize operation changes from subtraction to addition.
• Remember that subtracting a negative is like adding a positive.

Mastering negatives is crucial for arriving at the correct value, especially when dealing with expressions like this.