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Chapter 3: Problem 33

For equation, complete the solution. \(y=\frac{x}{4}+9 ;(16, \quad)\)

Short Answer

Expert verified
When \(x = 16\), \(y = 13\).

Step by step solution

01

Understand the Equation

We are given the linear equation \(y = \frac{x}{4} + 9\). This represents a line where \(y\) is the output, \(x\) is the input, and the constants \(\frac{1}{4}\) and 9 are the slope and y-intercept respectively. We need to find the value of \(y\) when \(x = 16\).
02

Substitute the Given Value

We need to substitute \(x = 16\) into the equation to find \(y\). Substitute \(x = 16\) in the equation: \(y = \frac{16}{4} + 9\).
03

Perform the Division

Begin by calculating \(\frac{16}{4}\). This equals 4. So the equation becomes \(y = 4 + 9\).
04

Add to Find \(y\)

Add the numbers: \(4 + 9 = 13\). Thus, \(y = 13\).

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Key Concepts

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

Slope
The slope of a linear equation tells you how steep the line is and the direction it moves. In our given equation, \(y = \frac{x}{4} + 9\), the slope is represented by \(\frac{1}{4}\). This means that for every unit increase in \(x\), the value of \(y\) increases by \( rac{1}{4}\) of that amount. It鈥檚 like climbing stairs 鈥� each step might be a different size depending on the slope.

Understanding the slope is crucial when working with linear equations because:
  • It determines the angle and direction of the line on a graph.
  • A positive slope, like \(\frac{1}{4}\), means the line goes upwards as you move from left to right.
  • A negative slope would mean the line moves downwards.
By knowing the slope, we can predict how changes in one variable affect the other. This concept is fundamental across various fields like physics and economics.
Y-Intercept
The y-intercept of a linear equation is the point where the line crosses the y-axis. It tells us the value of \(y\) when \(x\) is zero. In the equation \(y = \frac{x}{4} + 9\), the y-intercept is 9.

This means that if you were graphing this line,
  • It would cross the y-axis at the point (0, 9).
  • The y-intercept often represents a starting point in real-life scenarios, like the initial amount of money in a savings account.
Recognizing the y-intercept is essential because it provides a fixed reference point for the entire line. In equations, it's the constant term visible in the equation \(y = mx + b\), where \(b\) is the y-intercept. By identifying it, you can sketch the line or understand a system's initial conditions.
Substitution Method
The substitution method is a powerful algebraic tool used to find the values of unknown variables. In this exercise, we found \(y\) by substituting a specific value of \(x\) into the equation. With \(y = \frac{x}{4} + 9\) and \(x = 16\), substituting means replacing \(x\) with 16.

So, the step-by-step process is:
  • Insert \(x = 16\) into the equation: \(y = \frac{16}{4} + 9\).
  • Simplify the equation by performing the division: \(\frac{16}{4} = 4\).
  • Add 4 to 9, resulting in \(y = 13\).
This method is especially valuable when dealing with equations or systems of equations with more than one variable. By substituting known values, you systematically solve for the unknowns. It's widely applicable, from solving problems in textbooks to modeling real-world situations. By mastering the substitution method, you expand your problem-solving toolkit significantly.

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Most popular questions from this chapter

Find the slope and the y-intercept of the graph of each equation and graph it. See Examples 4 and 5. $$ 2 x+y=-6 $$

Determine whether the lines through each pair of points are parallel, perpendicular, or neither. See Example 8. \((-1,8)\) and \((-6,8)\) \((3,3)\) and \((3,7)\)

Why is \(y=m x+b\) called the slope-intercept form of the equation of a line?

In the report "Expenditures on Children by Families," the U.S. Department of Agriculture projected the yearly child-rearing expenditures on children from birth through age \(17 .\) For a child born in 2010 to a two-parent middle- income family, the report estimated annual expenditures of \(\$ 10,808\) when the child is 6 years old, and \(\$ 14,570\) when the child is 15 years old. a. Write two ordered pairs of the form (child's age, annual expenditure). b. Assume the relationship between the child's age \(a\) and the annual expenditures \(E\) is linear. Use your answers to part (a) to write an equation in slope-intercept form that models this relationship. c. What are the projected child-rearing expenses when the child is 17 years old?

Subscripts are used in other disciplines besides mathematics. In what disciplines are the following symbols used? a. \(\mathrm{H}_{2} \mathrm{O}\) and \(\mathrm{CO}_{2}\) b. \(\mathrm{C}_{7}\) and \(\mathrm{G}_{7}\) c. \(\mathrm{B}_{6}\) and \(\mathrm{B}_{12}\)
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