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Magazine Chapter 1: Problem 56
Make a table of values, and sketch the graph of the equation. $$3 x-5 y=30$$
Short Answer
Expert verified
Graph the line using points (0, -6), (5, -3), (10, 0), and (15, 3).
Step by step solution
01
Understand the Equation Format
The equation given is in standard form: \( 3x - 5y = 30 \). The goal is to rewrite it in a format that's easiest to graph by finding values for both \( x \) and \( y \).
02
Solve for y in terms of x
Rearrange the equation to solve for \( y \):\[ 3x - 5y = 30 \]Subtract \( 3x \) from both sides:\[ -5y = -3x + 30 \]Divide every term by \(-5\):\[ y = \frac{3}{5}x - 6 \]This is the slope-intercept form of the equation, \( y = mx + b \), where \( m = \frac{3}{5} \) and \( b = -6 \).
03
Choose Input Values for x
Choose several values for \( x \) to find corresponding \( y \) values. It is often useful to pick values that make calculations easy, such as multiples of 5, to work with the fraction \( \frac{3}{5} \).
04
Calculate Corresponding y Values
Using the equation \( y = \frac{3}{5}x - 6 \), find \( y \) for each chosen \( x \):- If \( x = 0 \), \( y = \frac{3}{5}(0) - 6 = -6 \)- If \( x = 5 \), \( y = \frac{3}{5}(5) - 6 = 3 - 6 = -3 \)- If \( x = 10 \), \( y = \frac{3}{5}(10) - 6 = 6 - 6 = 0 \)- If \( x = 15 \), \( y = \frac{3}{5}(15) - 6 = 9 - 6 = 3 \)
05
Create a Table of Values
List the \( x \) and \( y \) pairs in a table: \[\begin{array}{|c|c|}\hlinex & y \\hline0 & -6 \5 & -3 \10 & 0 \15 & 3 \\hline\end{array}\]
06
Plot the Points
Plot the points from the table on a coordinate grid:
- (0, -6)
- (5, -3)
- (10, 0)
- (15, 3)
These points should form a straight line.
07
Draw the Line
Connect the plotted points with a straight line to sketch the graph of the equation \(3x - 5y = 30\). Make sure the line extends across both axes for a clear visual representation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Standard Form Equations
Standard form is one way to write a linear equation. This format generally looks like this: \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers. In our equation, \( 3x - 5y = 30 \), \( A \) is 3, \( B \) is -5, and \( C \) is 30.
- Standard form equations provide a straightforward way to analyze and utilize equations.
- They are particularly helpful when dealing with equations that require finding intercepts.
Slope-Intercept Form
The slope-intercept form is a common way to express linear equations. It is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
- In our example, transforming \( 3x - 5y = 30 \) to \( y = \frac{3}{5}x - 6 \) allows us to easily identify the slope and y-intercept.
- The slope \( m = \frac{3}{5} \) tells us that for every 5 units we move horizontally, the line moves 3 units vertically.
- The y-intercept \( b = -6 \) indicates where the line crosses the y-axis.
Table of Values
Creating a table of values makes it easy to find and plot points for graphing linear equations.
- First, select values for \( x \) 鈥� choosing increments that simplify calculations, like 0, 5, 10, and 15, is often helpful.
- Substitute these \( x \)-values into the slope-intercept form to find the corresponding \( y \) values.
- When \( x = 0 \), \( y = -6 \)
- When \( x = 5 \), \( y = -3 \)
- When \( x = 10 \), \( y = 0 \)
- When \( x = 15 \), \( y = 3 \)
Coordinate Grid Plotting
Plotting on a coordinate grid is a key part of graphing linear equations. This grid consists of two axes: horizontal (x-axis) and vertical (y-axis).
The line not only visually represents the equation but also helps in understanding the relationship between \( x \) and \( y \) in the given equation.
- Start by marking the points from your table of values onto the grid.
- Each point corresponds to a pair \((x, y)\), such as \((0, -6)\), \((5, -3)\), etc.
- Ensure your points are plotted at their exact locations relative to both axes.
The line not only visually represents the equation but also helps in understanding the relationship between \( x \) and \( y \) in the given equation.
