Problem 98 - Elementary and Intermediate Algebra

Question

Complete each table. See Example 11. $$ \begin{array}{|c|c|} \hline g & {g^{2}-7 g+1} \\ \hline 0 & {} \\ \hline 7 & {} \\ \hline-10 & {} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

For \( g = 0 \) and \( g = 7 \), the expression evaluates to 1, and for \( g = -10 \), it evaluates to 171.

Step 1: Understanding the Table

The given table includes two columns. The first column represents the values of \( g \), and the second column represents the expression \( g^2 - 7g + 1 \). Our task is to compute the value of this expression for each provided value of \( g \).

Step 1: Evaluate the expression for \( g = 0 \)

Substitute \( g = 0 \) into the expression: \( g^2 - 7g + 1 \). This gives: \[ 0^2 - 7(0) + 1 = 0 - 0 + 1 = 1 \]Therefore, the value of the expression for \( g = 0 \) is 1.

Step 2: Evaluate the expression for \( g = 7 \)

Substitute \( g = 7 \) into the expression: \( g^2 - 7g + 1 \). This gives: \[ 7^2 - 7(7) + 1 = 49 - 49 + 1 = 1 \]Therefore, the value of the expression for \( g = 7 \) is 1.

Step 3: Evaluate the expression for \( g = -10 \)

Substitute \( g = -10 \) into the expression: \( g^2 - 7g + 1 \). This gives: \[ (-10)^2 - 7(-10) + 1 = 100 + 70 + 1 = 171 \]Therefore, the value of the expression for \( g = -10 \) is 171.

Step 5: Fill in the Table

Based on the computations, fill in the missing values in the table:\[\begin{array}{|c|c|}\hline g & g^2 - 7g + 1 \\hline 0 & 1 \\hline 7 & 1 \\hline -10 & 171 \\hline\end{array}\]

Key Concepts

Evaluating Expressions

Evaluating expressions is a key skill in algebra that involves substituting specific values for variables and simplifying the resulting expressions. When you evaluate an expression, you replace each variable in the expression with a number and perform the arithmetic operations according to the order of operations (parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right).
For example, consider the expression \(g^2 - 7g + 1\). To evaluate this expression for \(g = 0\), you substitute 0 in place of \(g\):

• Calculate \(0^2\), which equals 0.
• Multiply \(-7\) by 0, resulting in 0.
• Add 1 to the results: \(0 - 0 + 1\) equals 1.

This process gives you the simplified value of the expression, which is 1 when \(g = 0\). By practicing evaluating expressions, you become more comfortable with algebraic manipulation, which is critical for understanding more complex math topics.

Substitution in Algebra

Substitution is an essential method used in algebra to simplify expressions and solve equations. It involves replacing variables in an expression with values or other expressions.
In our example of the quadratic expression \(g^2 - 7g + 1\), substitution involves replacing \(g\) with different numbers from the table given in the exercise. For instance, substituting \(g = 7\):

• Calculate \(7^2\), which gives us 49.
• Multiply \(-7\) by 7, resulting in -49.
• Add 1 to these results, yielding \(49 - 49 + 1 = 1\).

Through substitution, we find that the value of the expression for \(g = 7\) is also 1. By understanding substitution, you can tackle a variety of algebra problems, making it a foundational tool for problem solving.

Problem Solving in Algebra

Problem solving in algebra often involves breaking down a problem into manageable steps, such as evaluating expressions and using substitution. These steps are crucial for finding solutions efficiently.
In our exercise, the problem asks us to determine the values of a quadratic expression for different values of \(g\). By systematically applying substitution to each given value of \(g\), we can solve the problem step by step. For example, with \(g = -10\):

• Compute \((-10)^2\), which is 100.
• Multiply \(-7\) by -10 to get 70.
• Add 1 to these numbers: \(100 + 70 + 1 = 171\).

Thus, for \(g = -10\), the value is 171. By breaking down the calculation steps, you effectively solve the problem and fill in the table accurately. Emphasizing structured problem-solving enhances your ability to tackle a wide range of algebraic challenges.