/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 52 Evaluate \(y=\frac{3}{4} x+9\) w... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 52

Evaluate \(y=\frac{3}{4} x+9\) when \(x=24\).

Short Answer

Expert verified
The value of \(y\) is 27 when \(x = 24\).

Step by step solution

01

Identify the Given Equation and Variable

The given equation is \(y = \frac{3}{4}x + 9\). We need to evaluate this equation for \(x = 24\).
02

Substitute the Value of \(x\)

Substitute \(x=24\) into the equation: \(y = \frac{3}{4}(24) + 9\).
03

Simplify the Multiplication

Calculate the value of the product \(\frac{3}{4}(24)\): \(\frac{3}{4} \times 24 = 18\).
04

Add the Constant Term

Now add the constant term to the result of the multiplication: \(y = 18 + 9\).
05

Compute the Final Value

Perform the addition: \(y = 27\).

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Key Concepts

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

Substitution
Substitution is a fundamental method in algebra where we replace one variable with a given value. In the given exercise, we substitute the value of \(x = 24\) in the equation \(y = \frac{3}{4}x + 9\). This helps us find the value of \(y\) for a specific \(x\). By replacing \(x\) with a number, we transform the equation into one that only involves numbers and constants, making it easier to solve.

Let's see how substitution was used in this exercise step-by-step:
  • Start with the original equation: \(y = \frac{3}{4}x + 9\)
  • Substitute the value of \(x\): \(x = 24\)
  • The equation becomes: \(y = \frac{3}{4}(24) + 9\)

The substitution step is crucial because it sets the stage for solving the equation by simplifying it through replacing variables with known values.
Linear Equations
Linear equations represent relationships where one variable depends directly on another, and they graph as straight lines. In this exercise, we have the linear equation \(y = \frac{3}{4}x + 9\). The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Breaking down our equation:
  • The slope \(m = \frac{3}{4}\), which tells us how steep the line is
  • The y-intercept \(b = 9\), which is where the line crosses the y-axis

When we substitute \(x = 24\) into the equation, we get:
  • \(y = \frac{3}{4}(24) + 9\)
  • This simplifies as we perform the arithmetic operations

Linear equations are simple yet powerful tools in algebra for depicting straight-line relationships between two variables.
Arithmetic Operations in Algebra
Arithmetic operations like addition, subtraction, multiplication, and division are fundamental elements in solving algebraic expressions. In our given exercise, we see these operations in action.

Here's a breakdown of the arithmetic steps involved:
  • Multiplication: Calculate \( \frac{3}{4} \times 24 \). The result is 18
  • Addition: Then add the constant term, 9, to the result of the multiplication: \( 18 + 9 = 27 \)
These operations simplify our equation step-by-step:
  • Start from \(y = \frac{3}{4}(24) + 9\)
  • Simplify multiplication: \(y = 18 + 9\)
  • Complete the addition: \(y = 27\)

Understanding these basic arithmetic operations is essential for evaluating and solving algebraic expressions successfully.

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Most popular questions from this chapter

Use the slope formula to find the slope of the line that passes through the points. \((-30,55) ;(-5,80)\)

Use the slope formula to find the slope of the line that passes through the points. \((-20,45) ;(-15,90)\)

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(x-y=8\)

For exercises 103-104, some learning preferences describe how you prefer to receive, think about, and learn new information. These preferences include visual learning, auditory learning, and kinesthetic learning. Many students use more than one of these categories as they learn mathematics. \- Visual learners prefer to see information. Although you definitely listen to your instructor, you also like to see the example on a white board or screen. You may be able to recall a process by visualizing it in your mind; you may learn better by organizing information in charts, tables, diagrams, or pictures. You may prefer the use of colored markers instead of just black. \- Auditory learners prefer to hear information. Although you definitely watch what your instructor is doing, you also like your instructor to explain things aloud as he or she works. You may find it difficult to take notes because you cannot concentrate enough on what is being said while you write. You may learn better if you have the chance to explain things to others. \- Kinesthetic learners prefer to do. You may find it difficult to sit still and just watch and listen; you want to be trying it out. You may find that you must take notes in order to learn. If you only watch and listen, you may understand the concept but not remember it after you leave the classroom. You often learn better if you can show others how to do things. Do you have a strong preference for visual, auditory, or kinesthetic learning?

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(-8 x+3 y=48\)
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