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Magazine Chapter 3: Problem 75
Graph each equation. $$ 3 x=-150-5 y $$
Short Answer
Expert verified
The line passes through the points (0, -30) and (5, -33) with a slope of -3/5.
Step by step solution
01
Rearrange the Equation
The given equation is \(3x = -150 - 5y\). To graph the equation, we need it in slope-intercept form \(y = mx + b\) or standard form \(Ax + By = C\). Let's first convert it to standard form by adding \(5y\) to both sides: \[3x + 5y = -150.\]
02
Identify Slope and Intercept
From the standard form \(3x + 5y = -150\), we can rearrange it into slope-intercept form to find the slope and y-intercept. Solve for \(y\) by subtracting \(3x\) from both sides and then dividing by \(5\): \[5y = -3x - 150\] \[y = -\frac{3}{5}x - 30.\] The slope \(m\) is \(-\frac{3}{5}\) and the y-intercept \(b\) is \(-30\).
03
Plot the Y-Intercept
Start graphing the equation by plotting the y-intercept on the coordinate plane. Since \(b = -30\), mark the point \( (0, -30) \) on the y-axis.
04
Use the Slope to Find Another Point
The slope \(-\frac{3}{5}\) means that for every 5 units you move to the right (positive direction of x), you move down 3 units (negative direction of y). From the y-intercept point \((0, -30)\), move 5 units right to \((5, -30)\), and then 3 units down to \((5, -33)\). Plot the point \((5, -33)\).
05
Draw the Line
With the points \((0, -30)\) and \((5, -33)\) plotted, draw a straight line through these points extending in both directions. This is the graph of the equation \(3x = -150 - 5y\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope-Intercept Form
The slope-intercept form of a linear equation is a compact way of writing an equation that makes it easy to identify its slope and the y-intercept. It is written as \( y = mx + b \). In this formula:
- \( m \) represents the slope of the line.
- \( b \) is the y-intercept.
Standard Form
The standard form of a linear equation is expressed as \( Ax + By = C \). In this format, \( A \), \( B \), and \( C \) are integers, and \( A \) should be a non-negative value. The standard form is beneficial because it can easily adapt to problems involving systems of equations and makes it simple to convert to other forms.
To graph a line from this form, you can either convert it to slope-intercept form or use intercepts.
To graph a line from this form, you can either convert it to slope-intercept form or use intercepts.
- Solve for the y-intercept by setting \( x = 0 \).
- Solve for the x-intercept by setting \( y = 0 \). This provides another point.
Y-Intercept
The y-intercept of a line is a crucial aspect to consider when graphing because it indicates where the line crosses the y-axis. It is denoted as \( b \) in the slope-intercept form \( y = mx + b \).
This point is represented by the coordinates \( (0, b) \) because at the y-intercept, the value of \( x \) is always zero. By marking this point on the graph, you establish a starting point for drawing the rest of the line.
This point is represented by the coordinates \( (0, b) \) because at the y-intercept, the value of \( x \) is always zero. By marking this point on the graph, you establish a starting point for drawing the rest of the line.
- The y-intercept provides an anchor point that helps visualize the line in relation to the axis.
- Knowing the y-intercept simplifies graphing, as you can then use the slope to determine other points on the line.
Slope
The slope of a line is a measure of its steepness and direction. It helps in understanding how the y-value of a line changes for every change in the x-value. Slope is noted as \( m \) in the slope-intercept form of a line \( y = mx + b \).
- If \( m \) is positive, the line rises as it moves from left to right.
- If \( m \) is negative, the line falls as it moves from left to right.
