Problem 93 - Algebra 1: Concepts and Skills

Question

If \(a^{2}=36\) and \(b^{2}=49,\) choose the greatest possible value for the expression \(b-a\) f. -13 g. -1 h. 1 j. 13

Step-by-Step Solution

Answer

The greatest possible value for the expression \(b - a\) is j. 13

Step 1: Find the square roots

For \(a^{2}=36\) and \(b^{2}=49\), find the square roots for these numbers. The square root of 36 is \(\pm6\) (meaning \(a\) could either be 6 or -6) and the square root of 49 is \(\pm7\) (meaning \(b\) could either be 7 or -7).

Step 2: Subtract the value of a from b

To get the maximum possible result for \(b - a\), we need to subtract the smallest possible value of \(a\) from the largest possible value of \(b\). That would be \(7 - (-6)\) which is equal to 13.

Step 3: Select the correct answer

Having calculated the value of \(b - a\), we can now choose the correct answer from the given options. The maximum possible value for \(b - a\) is 13, which corresponds to answer choice j. 13

Key Concepts

Square Roots

Square roots are a fundamental topic in algebra, essential for solving equations involving variables squared. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, for the equation \(a^2 = 36\), the square roots of 36 are 6 and -6, because \(6 \times 6 = 36\) and \((-6) \times (-6) = 36\). Each positive number has two square roots: a positive and a negative. In solving algebraic equations, it’s crucial to consider both roots unless specified otherwise.

• Be aware that the square root symbol \(\sqrt{}\) typically refers to the principal (non-negative) square root.
• When solving equations, examine the equation to determine if both positive and negative roots are relevant to the solution.

Square roots help unlock the values hidden within squared variables and lay the groundwork for more complex algebraic concepts.

Solving Equations

Solving equations is a central task in algebra. This involves finding the value of the variable that makes the equation true. Let’s explore the purpose through the example of the equation \(a^2 = 36\). To solve this equation, find the values of \(a\) that satisfy it by considering its square roots: 6 and -6.
Subtract these solutions as per the problem to determine \(b - a\). If \(b=7\) and \(a=-6\), the calculation becomes simple:

• Perform \(b - a\): 7 - (-6) = 7 + 6 = 13.
• Practice such steps to ensure a firm understanding of basic operations and relationships between variables.

By following logical steps and operations, solving equations enables you to find meaningful solutions to algebraic expressions, unraveling values behind variables.

Inequalities

Inequalities differ from equations as they describe a range of possible answers rather than a specific solution. They are crucial in problems involving comparisons and orderings, such as finding maximum or minimum values. While not prominent in the original exercise, understanding inequalities helps when determining constraints on variables, such as seeking the greatest value for \(b-a\).

• Inequalities often include symbols such as ">", "<", "\(\geq\)", and "\(\leq\)".
• Manipulate inequalities much like equations but be cautious: multiplying or dividing by a negative number reverses the inequality sign.

Developing proficiency with inequalities alongside traditional equations enhances skills in analyzing and solving complex algebraic problems.