Problem 106 - Beginning and Intermediate Algebra

Question

If \(x=-5\) and \(y=-3,\) evaluate each expression. $$ x^{2}-2 y^{2} $$

Step-by-Step Solution

Answer

7

Step 1: Substitute the Values

Substitute the given values of \(x\) and \(y\) into the expression \(x^{2} - 2y^{2}\). This gives \((-5)^{2} - 2(-3)^{2}\).

Step 2: Calculate Each Squared Term

Calculate \((-5)^{2}\) and \((-3)^{2}\). We have \((-5)^{2} = 25\) and \((-3)^{2} = 9\).

Step 3: Multiply and Simplify

Multiply \(2\) by \(9\), resulting in \( 2 imes 9 = 18\).

Step 4: Evaluate the Expression

Now, substitute the squared values and multiply result into the expression: \(25 - 18\).

Step 5: Perform the Final Calculation

Calculate the result of \(25 - 18\), which is \(7\).

Key Concepts

Substitution

Substitution is one of the most fundamental and useful techniques in mathematics. It involves replacing variables in an equation or expression with their given values. This allows us to evaluate and find a solution.To understand substitution fully:

• Identify the variables in the expression. In this case, they are \(x\) and \(y\).
• Each variable is replaced with its given value. For example, substituting \(x = -5\) and \(y = -3\) into the equation \(x^{2} - 2y^{2}\) transforms it into \((-5)^{2} - 2(-3)^{2}\).

Substitution is like replacing letters in a word puzzle with known letters to find the solution. With each variable replaced, you are ready to simplify and solve the expression.

Exponents

Exponents refer to the mathematical operation that involves raising a number to a certain power. This indicates how many times that number should be multiplied by itself.In our given expression, we have both \((-5)^2\) and \((-3)^2\). This means:

• \((-5)^2\) is calculated as \((-5) \times (-5) = 25\).
• \((-3)^2\) is calculated as \((-3) \times (-3) = 9\).

It's important to understand that squaring a negative number results in a positive result. So, in our case, both results are positive numbers. Exponents make it easy to express the repeated multiplication of the same number.

Simplification

Simplification involves reducing an expression to its most basic form. This process usually makes the problem easier to understand and solve.For the equation \((-5)^{2} - 2(-3)^{2}\):

• We first calculate \((-5)^2\) to simplify it to \(25\).
• Next, calculate \((-3)^2\) to get \(9\). Multiply this by \(2\) which yields \(18\).
• The expression now reads \(25 - 18\).

Simplification reduces unnecessary complexity, making it straightforward to find a direct solution step.

Step-by-Step Solutions

Solving mathematical problems systematically helps avoid errors and confusion. Step-by-step solutions guide you through the process of solving an expression or equation in logical sequences.The steps for evaluating \(x^{2} - 2y^{2}\) were:

• Step 1: Substitution of given values into the expression.
• Step 2: Calculate each part, such as the squared terms.
• Step 3: Handle any multiplication within the expression.
• Step 4: Substitute back into the simplified expression.
• Step 5: Perform the final arithmetic calculation.

Following each step ensures all parts of the expression are correctly evaluated, making the final outcome accurate.