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Magazine Chapter 10: Problem 2
Draw the graph of each equation. Name any intercepts. $$3 x+5 y=15$$
Short Answer
Expert verified
X-intercept: (5, 0); Y-intercept: (0, 3). Graph is a line through these points.
Step by step solution
01
Identify Intercepts
To sketch the graph of the equation \(3x + 5y = 15\), we first find the x-intercept and y-intercept. The x-intercept occurs where \(y = 0\), while the y-intercept occurs where \(x = 0\).
02
Find the X-Intercept
Set \(y = 0\) in the equation \(3x + 5y = 15\). \[3x + 5(0) = 15 \3x = 15 \x = \frac{15}{3} = 5\]So, the x-intercept is at the point \((5, 0)\).
03
Find the Y-Intercept
Set \(x = 0\) in the equation \(3x + 5y = 15\). \[3(0) + 5y = 15 \5y = 15 \y = \frac{15}{5} = 3\]So, the y-intercept is at the point \((0, 3)\).
04
Draw the Graph
Plot the intercepts \((5, 0)\) and \((0, 3)\) on the coordinate plane. Draw a straight line through these points to represent the equation \(3x + 5y = 15\).
05
Verify and Name Intercepts
Ensure the line passes through both points. The intercepts are correctly identified as the x-intercept \((5, 0)\) and the y-intercept \((0, 3)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Intercepts
Graphing intercepts is a fundamental step in plotting a linear equation. Intercepts provide the points where a line crosses the x- and y-axes on the coordinate plane. To find them, we set each variable to zero, one at a time.
- X-Intercept: By setting y to zero in the equation, you can find where the line crosses the x-axis. This is where the line meets the horizontal axis.
- Y-Intercept: By setting x to zero in the equation, you can determine where the line crosses the y-axis, matching the line with the vertical axis.
Coordinate Plane
The coordinate plane is a two-dimensional surface formed by the intersection of a horizontal line (x-axis) and a vertical line (y-axis). These axes divide the plane into four quadrants, providing a framework for graphing equations.
- Origin: The point (0,0) where the x-axis and y-axis intersect. It is the starting point for measuring distances along the axes.
- Quadrants: The four sections of the coordinate plane, typically numbered counterclockwise. Each has its unique pair of positive and negative values.
Algebraic Graphing
Algebraic graphing involves representing algebraic equations on a graph. It is a visual method to understand how equations behave and relate to one another.
To graph an equation like the one given, follow these steps:
To graph an equation like the one given, follow these steps:
- Calculate intercepts. This provides clear points of reference on the graph.
- Plot the intercepts on the coordinate plane.
- Draw a line through these plotted points. For linear equations, this line will extend infinitely in both directions.
Equation of a Line
The equation of a line is a mathematical expression that describes a straight line on the coordinate plane. It can be presented in various forms, with each providing insights into different aspects of the line.
- Standard Form: As seen in this problem, the equation is in standard form, represented as Ax + By = C. It's a straightforward way to express any linear equation. From here, intercepts can be efficiently calculated.
- Slope-Intercept Form: Another common form is y = mx + b. Here, "m" represents the slope, showing how steep the line is, and "b" represents the y-intercept, indicating where the line crosses the y-axis.
