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Chapter 3: Problem 110

Will help you prepare for the material covered in the next section. a. If \(y=k x^{2},\) find the value of \(k\) using \(x=2\) and \(y=64\) b. Substitute the value for \(k\) into \(y=k x^{2}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=5\)

Short Answer

Expert verified
a. The value of \(k\) is 16. b. The resulting equation is \(y=16 x^{2}\). c. When \(x=5\), \(y\) equals 400.

Step by step solution

01

Solve for \(k\)

To solve for \(k\), we are given that \(x=2\) and \(y=64\). Substituting these values into the equation \(y=k x^{2}\), we get \(64=k(2)^{2}\). This simplifies to \(64=4k\). To isolate \(k\), we then divide both sides of the equation by 4. Doing that gives us \(k=64/4=16\).
02

Substitution of \(k\) into Equation

Now substitute \(k=16\) into the original equation \(y=k x^{2}\). This yields \(y=16 x^{2}\).
03

Calculation of \(y\) Value

Use the equation from the previous step to find the value of \(y\) when \(x=5\). Substituting \(x=5\) into the equation \(y=16 x^{2}\) gives us \(y=16(5)^{2}\), which simplifies to \(y=16(25)=400\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving for a Variable
Finding the value of an unknown variable in an equation can seem tough at first. But once you understand the process, it becomes much easier. Let's explore how to solve for the variable \(k\) using the equation \(y = kx^2\). This involves determining the specific value of \(k\) that makes the equation true given certain conditions.

First, we substitute the known values into the equation. In our example, if \(x = 2\) and \(y = 64\), substituting them into \(y = kx^2\) gives \(64 = k(2)^2\).

The aim is to have \(k\) by itself on one side of the equation. To do that, simplify the right side of the equation: \((2)^2 = 4\). Now, the equation becomes \(64 = 4k\).

To isolate \(k\), divide both sides of the equation by 4, which gives you \(k = \frac{64}{4} = 16\). Simple, right? This tells us that when \(x = 2\), the value of \(k\) is 16 in order for \(y\) to be 64.
Substitution Method
Substitution is a technique where we replace a variable with a given value or another expression. It's like swapping one piece of the puzzle for another that fits perfectly.

Once you've solved for a variable, it can be substituted back into the original equation. In our scenario, after determining that \(k = 16\), we plug this value back into the equation \(y = kx^2\).

By substituting \(k = 16\) into the equation, it becomes \(y = 16x^2\).

This new equation can be used to find different possible values of \(y\) depending on the given \(x\). Substitution is handy for turning a complex equation into a simpler one. It helps to uncover new insights about how changes in one part of the equation affect the rest.
Equation Manipulation
Equation manipulation is the art of rearranging equations to get the information you need. To find \(y\) when \(x = 5\) using the equation \(y = 16x^2\), we rearrange the expression to plug in the new \(x\).

Think of it as unlocking parts of the equation to reach a solution. Substitute \(x = 5\) into \(y = 16x^2\), resulting in \(y = 16(5)^2\).

Calculate \((5)^2\) first, which equals 25, and then do the multiplication \(16 \times 25\). The result is \(y = 400\).

By manipulating the equation, you find that when \(x = 5\), \(y\) equals 400. This skill is essential in algebra and beyond because it allows you to make predictions and solve problems using different variables and values. Equation manipulation lets you explore endless possibilities within mathematical relationships.

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