Problem 27 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ 5 x-4 y-7 y+y-7 x, \text { if } x=1 \text { and } y=-2 $$

Step-by-Step Solution

Answer

18

Step 1: Substitute Values into Expression

First, substitute the given values of \( x = 1 \) and \( y = -2 \) into the expression \( 5x - 4y - 7y + y - 7x \). This gives us:\[5(1) - 4(-2) - 7(-2) + (-2) - 7(1)\]

Step 2: Simplify Each Term

Simplify each term in the expression by performing the multiplication:- \( 5(1) = 5 \)- \( -4(-2) = 8 \)- \( -7(-2) = 14 \)- \( -2 \) remains the same- \( -7(1) = -7 \)So, the expression becomes:\[5 + 8 + 14 - 2 - 7\]

Step 3: Combine the Terms

Add and subtract the terms step-by-step:1. Combine \(5 + 8 = 13\)2. Add \(13 + 14 = 27\)3. Subtract \(27 - 2 = 25\)4. Subtract \(25 - 7 = 18\)The value of the expression is 18.

Key Concepts

Substitution Method

The substitution method in algebra involves replacing variables in an expression with given numerical values. This technique is quite helpful to determine the specific outcome of an expression for certain values of variables.

• First, identify the variables in your algebraic expression. In our example, the variables are \(x\) and \(y\).
• Next, substitute each variable with its provided numerical value. For \(x = 1\), replace each \(x\) in the expression with 1. Similarly, substitute \(y = -2\) where you see a \(y\).

By systematically doing this replacement, such as turning \(5x\) into \(5(1)\), you transform the expression into numerical terms where further algebraic manipulation can continue. Essentially, you're swapping out variables for numbers to simplify calculations.

Simplifying Expressions

Once you've substituted the variables with their respective values, the next step is to simplify the expression. Simplifying involves carrying out arithmetic operations like multiplication and combining values properly to reduce the expression to its simplest form.

• Evaluate individual terms. For instance, \(5(1)\) becomes 5.
• Pay attention to signs. A negative sign can affect the outcome of your operations, like \(-4(-2)\) resulting in a positive 8.
• Ensure all similar operations are done to each part of the expression.

Properly simplifying an expression helps in minimizing processing errors and getting a clearer picture of what the expression represents numerically.

Combining Like Terms

After simplifying the individual terms, the final step is to combine like terms to reach a single numerical value.

• Add or subtract terms that have been simplified to reach a more condensed form.
• Keep in mind any existing arithmetic signs from previous steps as they impact the result.
• Following the order of operations correctly helps in accurately reaching the final expression value.

For example, once simplification is complete, you would combine like terms: start with sums \(5 + 8\) progressing through until final subtraction \(25 - 7 = 18\). This step ensures that the expression is finally reduced to a single, accurate value.