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Chapter 3: Problem 62

Graph each equation. \(3 y=4.5 \quad\) (Hint: Solve for \(y\) first.

Short Answer

Expert verified
Graphing the horizontal line \(y = 1.5\).

Step by step solution

01

Solve for y

First, solve the equation for \(y\). The given equation is \(3y = 4.5\). To isolate \(y\), divide both sides of the equation by 3: \[ y = \frac{4.5}{3} \] Calculating the division gives \(y = 1.5\).
02

Identify the Equation of the Line

After solving for \(y\), you have \(y = 1.5\). This is a horizontal line equation, indicating that \(y\) is constant irrespective of the \(x\) value.
03

Graph the Line

On a coordinate plane, mark a horizontal line at \(y = 1.5\). This line will be parallel to the \(x\)-axis and pass through the point (0, 1.5). Every point on this line will have the form \((x, 1.5)\), where \(x\) can be any real number.

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Key Concepts

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

Horizontal Lines
A horizontal line on the coordinate plane is a special type of line that has unique properties. It doesn't matter what the value of \( x \) is; the value of \( y \) remains the same across the entire line. This means that every point on a horizontal line will have a consistent \( y \)-value. In the given equation, once we solve for \( y \), it tells us that the line is horizontal at \( y = 1.5 \).

Here's what makes horizontal lines interesting:
  • They have a zero slope, meaning they don鈥檛 rise or fall as you move along the \( x \)-axis.
  • They are represented by equations in the form \( y = c \), where \( c \) is a constant.
  • They run parallel to the \( x \)-axis of the coordinate plane.
Visualizing this, imagine a level path on a hill鈥攏o ups and downs! It's important to pin down these basics since horizontal lines often pop up in algebra and coordinate geometry.
Coordinate Plane
The coordinate plane is like a map for graphing equations, providing a grid with two axes that intersect at a point called the origin.

Here鈥檚 a breakdown of its main features:
  • The horizontal axis is known as the \( x \)-axis.
  • The vertical axis is known as the \( y \)-axis.
  • The point where both axes meet is called the origin, represented by (0, 0).

Graphing on this plane allows you to accurately represent mathematical equations visually. When graphing a line, such as the horizontal line \( y = 1.5 \), you draw across the plane where \( y \) has a constant value of 1.5. You'll notice this line stretches left to right without tilting up or down, cutting through the y-axis at the point (0, 1.5). Mastering the coordinate plane is crucial for understanding how equations translate into visual graphs.
Solving for y
Solving for \( y \) is a fundamental skill in algebra as it often reveals more about the type of line an equation will graph.

In the exercise, you began with the equation \( 3y = 4.5 \). Here鈥檚 how you solve it step-by-step:
  • You start by isolating \( y \) on one side of the equation. Divide both sides of the equation by the coefficient of \( y \), which is 3 in this case.
  • This results in \( y = \frac{4.5}{3} \).
  • Upon performing the division, you find \( y = 1.5 \).

By isolating \( y \), you've translated the equation into a form that directly tells you the line is horizontal because it's in the form \( y = c \). Solving for \( y \) this way is a vital step in understanding graphing concepts, helping you map out the line accurately on a coordinate plane.

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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. $$ 2 x+3 y=9 $$

Lawn Sprinklers. The function \(A(r)=\pi r^{2}\) can be used to determine the area in square feet that will be watered by a rotating sprinkler that sprays out a stream of water. Find \(A(5)\) and \(A(20) .\) Round to the nearest tenth. CAN'T COPY THE IMAGE

For each pair of equations, determine whether their graphs are parallel, perpendicular, or neither. See Example 6 $$ \begin{aligned} &x=9\\\ &y=8 \end{aligned} $$

Use the concept of slope to determine whether \(A(-50,-10), B(20,0),\) and \(C(34,2)\) all lie on the same straight line.

If the graph of \(y=m x+b\) passes through quadrants I, II, and IV, what can be known about the constants \(m\) and \(b ?\)
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