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Chapter 2: Problem 24

For exercises 1-28, solve the equation for \(y\). Write the equation to match the pattern \(y=m x+b\). $$ 3 y=10 x $$

Short Answer

Expert verified
The equation is \(y = \frac{10}{3}x\).

Step by step solution

01

- Identify the given equation

The equation provided is \(3y = 10x\). This needs to be rearranged into the form \(y = mx + b\).
02

- Isolate the variable y

To solve for \(y\), divide both sides of the equation by 3.
03

- Divide both sides by 3

Dividing both sides by 3, the equation becomes \(y = \frac{10}{3}x\).
04

- Write the final form

The equation \(y = \frac{10}{3}x\) is in the form \(y = mx + b\) with \(m = \frac{10}{3}\) and \(b = 0\).

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Key Concepts

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

linear equations
A linear equation is a type of equation where the highest power of the variable is one. These equations form a straight line when plotted on a graph. The standard ways to write a linear equation are in the forms of \(y = mx + b\) or \(Ax + By = C\). Each term in a linear equation can be either a constant or a product of a constant and the variable. Writing equations in linear form allows us to understand the relationship between the variables and predict values more effectively.
slope-intercept form
The slope-intercept form \(y = mx + b\) is a way of writing linear equations clearly showing the slope and y-intercept of the line. Here, 'm' represents the slope, which shows the steepness and direction of the line. 'b' represents the y-intercept, or the point where the line crosses the y-axis. For the equation \(3y = 10x\), we converted it to \(y = \frac{10}{3}x\). In this equation, the slope 'm' is \(\frac{10}{3}\), and the y-intercept 'b' is 0.
solving for y
To solve for 'y' means isolating the variable 'y' on one side of the equation. This process usually involves algebraic manipulation. Since our initial equation was \(3y = 10x\), we divided both sides by 3 to isolate 'y', resulting in the equation \(y = \frac{10}{3}x\). By isolating 'y', we can easily understand how 'y' changes with 'x'.
algebraic manipulation
Algebraic manipulation involves operations that simplify or rearrange equations to make them more understandable or solvable. Common operations include addition, subtraction, multiplication, division, and distribution. In our example, to isolate 'y' in the equation \(3y = 10x\), we performed algebraic manipulation by dividing both sides of the equation by 3. This step transformed the equation to \(y = \frac{10}{3}x\), achieving a form that is easier to work with and analyze.

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Most popular questions from this chapter

For exercises 77-78, \(V=\frac{L W T}{144 \text { in. }^{3}}\) where \(V\) is the volume of lumber in board feet, \(L\) is the length, \(W\) is the width, and \(T\) is the thickness. The units of length, width, and thickness are inches. a. Solve \(V=\frac{L W T}{144 \text { in. }^{3}}\) for \(L\). Include the units of measurement. The thickness of each board in a pile is \(1.5\) in., and the width of each board is \(7.25 \mathrm{in}\). The volume of the boards is 348 board feet. Find the total length of the boards. .. Change this length into feet.

In Sept. 2010, \(1.4\) million people received cash assistance from the California Work Opportunity and Responsibility to Kids (CalWORKS) program. More than three out of four of these recipients were children. Find the minimum number of recipients who were children. (Source: www.lafla.org, Jan 2011)

Each of the three sides of an equilateral triangle is the same length. If \(P\) is the perimeter of the triangle and \(L\) is the length of a side, write a formula in \(P\) and \(L\) for the perimeter of an equilateral triangle and solve this formula for \(L\).

For exercises 37-52, (a) solve. (b) use a number line graph to represent the solution. (c) check the direction of the inequality sign. $$ 7 x+3<52 $$

Find the enrollment at Oregon State University before the increase. Round to the nearest whole number. Oregon State University saw the biggest growth, a \(5.3\) percent jump of 1,302 students. Portland State University grew by only \(1.5\) percent, but remains the state's largest university with 28,958 students. (Source: www.oregonlive .com, Nov. 10. 2011)
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