/*! 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 37 Graph the given equation. $$ ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 37

Graph the given equation. $$ 2 x=3 y $$

Short Answer

Expert verified
The graph of the given equation \(2x=3y\) can be represented by a line passing through the points (0, 0) and (3, 2) with a slope of 2/3 and a y-intercept of 0. In slope-intercept form, the equation is \(y = \frac{2}{3}x\).

Step by step solution

01

Choose two points on the line

We'll begin by choosing two values for x, and using the equation to find the corresponding values of y. We can use any values of x; for simplicity, let's choose x = 0 and x = 3.
02

Find the corresponding y values

We will now substitute our chosen x-values into the equation and solve for the corresponding y-values. For x = 0: \(2(0) = 3y\) \(0 = 3y\) \(y = 0\) For x = 3: \(2(3) = 3y\) \(6 = 3y\) \(y = 2\) Now we have two points: (0, 0) and (3, 2).
03

Draw the line

With our two points, (0, 0) and (3, 2), draw the line that passes through these points. This line represents the graph of the equation 2x = 3y.
04

Convert the given equation to slope-intercept form

To verify our graph, we'll convert the equation 2x = 3y to slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. Divide both sides of the equation by 3: \(\frac{2x}{3} = y\) Now we have the equation in the slope-intercept form, with a slope of 2/3 and a y-intercept of 0: \(y = \frac{2}{3}x\) With the equation in slope-intercept form, we can see that our graph is correct, as it passes through the points (0, 0) and (3, 2) with a slope of 2/3.

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Key Concepts

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

Slope-Intercept Form
Graphing a linear equation can be made easier by converting it to slope-intercept form, which is given by the equation:
\[ y = mx + b \]
where \( m \) represents the slope of the line and \( b \) denotes the y-intercept, which is where the line crosses the y-axis. In the given exercise, converting the equation \( 2x = 3y \) to slope-intercept form resulted in \( y = \frac{2}{3}x \). This indicates the line has a slope of \( \frac{2}{3} \) and crosses the y-axis at 0 (the origin). Understanding and using the slope-intercept form provides a straightforward way to plot a linear equation on the coordinate plane by identifying these two key characteristics of the line.
Linear Equations
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be graphed on a coordinate plane as straight lines. They are in the form of \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. The equation \( 2x = 3y \) from our problem is a linear equation and represents a straight line. By finding two points that satisfy the equation, which in our case are (0,0) and (3,2), you can plot the line by drawing a straight line through these points. This is a crucial step in visualizing the solution to a linear equation and understanding its behavior.
Coordinate Plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface consisting of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is the origin, marked as (0,0). Every point on the plane is defined by an ordered pair of numbers known as coordinates (x,y). When graphing linear equations, we locate points on the coordinate plane whose coordinates satisfy the equation. For example, in the exercise, the points we identified were (0,0) and (3,2). Plotting these points and drawing a line through them on the coordinate plane graphs the linear equation, providing a visual representation of all the possible solutions to the equation.

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Most popular questions from this chapter

Use technology to compute the sum-ofsquares error (SSE) for the given set of data and linear models. Indicate which linear model gives the better fit. $$ (0,1),(1,1),(2,2) ; \quad \text { a. } y=0.4 x+1.1 \quad \text { b. } y=0.5 x+0.9 $$

Find a linear equation whose graph is the straight line with the given properties. Through \((6,6)\) and parallel to the line \(x+y=4\)

Compute the sum-of-squares error \((S S E)\) by hand for the given set of data and linear model. $$ (0,-1),(1,3),(4,6),(5,0) ; \quad y=-x+2 $$

It cost Apple approximately \(\$ 800\) to manufacture 5 30-gigabyte video iPods and \(\$ 3,700\) to manufacture \(25 .^{*}\) Obtain the corresponding linear cost function. What was the cost to manufacture each additional iPod? Use the cost function to estimate the cost of manufacturing 100 iPods.

In the Fahrenheit temperature scale, water freezes at \(32^{\circ} \mathrm{F}\) and boils at \(212^{\circ} \mathrm{F}\). In the Celsius scale, water freezes at \(0^{\circ} \mathrm{C}\) and boils at \(100^{\circ} \mathrm{C}\). Assuming that the Fahrenheit temperature \(F\) and the Celsius temperature \(C\) are related by a linear equation, find \(F\) in terms of \(C\). Use your equation to find the Fahrenheit temperatures corresponding to \(30^{\circ} \mathrm{C}, 22^{\circ} \mathrm{C},-10^{\circ} \mathrm{C}\), and \(-14^{\circ} \mathrm{C}\), to the nearest degree.
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