Problem 42 - Elementary Algebra Within Reach

Question

In Exercises 37-44, evaluate the algebraic expression for the given values of the variable(s). \(10 u-3 v\) (a) \(u=3, v=10\) (b) \(u=-2, v=\frac{4}{7}\)

Step-by-Step Solution

Answer

The short answers for the given values of the variables are: (a) \(0\), and (b) \(- \frac{152}{7}\).

Step 1 - Substitution (a)

Firstly, we substitute the given values \(u=3\) and \(v=10\) into the expression, giving us: \(10u - 3v = 10(3) - 3(10)\)

Step 2 - Arithmetic (a)

Then, we simplify the expression: that gives us \(10(3) - 3(10) = 30 - 30 = 0\)

Step 3 - Substitution (b)

Next, substitute \(u=-2\) and \(v= \frac{4}{7}\) into the expression: \(10u - 3v = 10(-2) - 3(\frac{4}{7})\)

Step 4 - Arithmetic (b)

Finally, simplify this expression to get: \(10(-2) - 3(\frac{4}{7}) = -20 - \frac{12}{7} = \frac{-140}{7} - \frac{12}{7} = - \frac{152}{7}\)

Key Concepts

Substitution in Algebra

When dealing with algebraic expressions, one of the first skills you'll need to master is substitution. This involves replacing variables, such as 'u' and 'v', with the numbers or expressions they represent. In the context of our original exercise, the expression \(10 u-3 v\) becomes something we can work with once we know the values of 'u' and 'v'.

For example, given that \(u=3\) and \(v=10\), we directly replace 'u' with 3 and 'v' with 10 to evaluate the expression, transforming \(10 u-3 v\) into \(10(3)-3(10)\). Substitution is a fundamental technique that lays the groundwork for further steps in solving algebraic problems. Always ensure that you have substituted correctly before proceeding, as any mistake here will affect the entire solution.

Arithmetic Operations

The backbone of algebra lies in the four basic arithmetic operations: addition, subtraction, multiplication, and division. Once the substitution is done, as we saw with \(10(3)-3(10)\) from the previous step, we then apply these arithmetic operations to simplify the expression.

Let's bring our focus to multiplication and subtraction for our given example. After substituting, we multiply the coefficients (10 and -3) by their respective variable values (3 and 10 for part (a)). The resulting operation \(30 - 30\) involves subtraction, another arithmetic operation, which simplifies down to 0. Understanding these operations and applying them correctly is essential to evaluating algebraic expressions accurately, ensuring that the simplification process is conducted with ease.

Simplifying Expressions

The final goal with algebraic expressions is to simplify them as much as possible. This means performing all the operations and reducing the expression down to its simplest form. After the substitution step gives us \(10(-2)-3(\frac{4}{7}))\), we then perform the multiplication which leads to \( -20 - \frac{12}{7} \).

The last step involves adding or subtracting the numbers like in our example, where we are subtracting fractions with a common denominator. This results in \(\frac{-140}{7} - \frac{12}{7} = - \frac{152}{7}\), a simplified form of the original expression. Simplification helps in understanding the core value of the expression and makes it more readable and easier to use in further mathematical or real-life applications.