Problem 68
Question
Use intercepts to graph each equation. \(6 x-9 y-18-0\)
Step-by-Step Solution
Verified Answer
The x-intercept is 3 and the y-intercept is -2. The graph of the equation \(6x - 9y = 18\) passes through these intercept points (3,0) and (0,-2).
1Step 1: Find the x-intercept
The x-intercept is the value of x when y equals to zero. In order to find the x-intercept, set y equal to zero and solve the equation for x. The given equation is \(6x - 9y = 18\). If we set \(y = 0\), the equation becomes \(6x - 0 = 18\). Thus, solving for x yields \(x = \frac{18}{6} = 3\). As a result, the x-intercept is 3.
2Step 2: Find the y-intercept
The y-intercept is the value of y when x equals to zero. To find the y-intercept, set x equal to zero and solve the equation for y. Using the equation \(6x - 9y = 18\), if we let \(x = 0\), the equation simplifies to \(-9y = 18\). Solving this equation yields \(y = \frac{18}{-9} = -2\). Thus, the y-intercept is -2.
3Step 3: Graph the equation using intercepts
Plot the points of the x-intercept (3,0) and y-intercept (0,-2) on a graph. Draw a line that passes through these points, and this would represent the equation \(6x - 9y = 18\) on a graph.
Key Concepts
Understanding the x-interceptUnpacking the y-interceptSolving Equations for InterceptsThe Coordinate Plane BasicsPlotting Points and Drawing Lines
Understanding the x-intercept
The x-intercept of a line is the point where the line crosses the x-axis on a coordinate plane. This occurs when the value of y is zero in the equation of the line. To find the x-intercept:
For example, in the equation provided, \(6x - 9y = 18\), setting \(y = 0\) simplifies the equation to \(6x = 18\). Solving for x gives us \(x = 3\). Therefore, the x-intercept is 3, or the point \((3, 0)\) on the graph.
Knowing the x-intercept helps us understand where the graph of the line intersects the x-axis.
- Set y equal to zero in the equation.
- Solve the equation for x.
For example, in the equation provided, \(6x - 9y = 18\), setting \(y = 0\) simplifies the equation to \(6x = 18\). Solving for x gives us \(x = 3\). Therefore, the x-intercept is 3, or the point \((3, 0)\) on the graph.
Knowing the x-intercept helps us understand where the graph of the line intersects the x-axis.
Unpacking the y-intercept
The y-intercept is where a line crosses the y-axis on a coordinate plane. This point is found by setting x to zero in the equation. Similar to finding the x-intercept, this step provides another key point on our graph.
In our equation, \(6x - 9y = 18\), set \(x = 0\). The equation simplifies to \(-9y = 18\), so solving for y gives us \(y = -2\). Hence, the y-intercept is -2, represented by the coordinate \((0, -2)\).
Finding the y-intercept helps us visualize where the line touches the y-axis, offering a critical reference point for graphing.
In our equation, \(6x - 9y = 18\), set \(x = 0\). The equation simplifies to \(-9y = 18\), so solving for y gives us \(y = -2\). Hence, the y-intercept is -2, represented by the coordinate \((0, -2)\).
Finding the y-intercept helps us visualize where the line touches the y-axis, offering a critical reference point for graphing.
Solving Equations for Intercepts
Solving equations is a vital process that allows us to find coordinates for intercepts. This process involves isolating the variable on one side to find specific values for x and y.
This method can highlight key points on a graph, like intercepts, which guide you in plotting the linear equation accurately.
For instance,
Practicing these steps ensures a strong foundation in graphing and interpreting linear relationships.
This method can highlight key points on a graph, like intercepts, which guide you in plotting the linear equation accurately.
For instance,
- Set variable values (such as y or x to zero) in \(6x - 9y = 18\)
- Solve for the other variable by simple arithmetic operations, like addition, subtraction, division, or multiplication.
Practicing these steps ensures a strong foundation in graphing and interpreting linear relationships.
The Coordinate Plane Basics
The coordinate plane is a two-dimensional surface used to graph equations. It consists of two perpendicular number lines called axes: the x-axis (horizontal) and the y-axis (vertical).
Each point on the plane is identified by a pair of numbers, known as coordinates, written as \((x, y)\).
Understanding how to navigate the coordinate plane is essential for plotting points and graphing equations accurately.
Each point on the plane is identified by a pair of numbers, known as coordinates, written as \((x, y)\).
- The center of the axes is the origin, marked as \((0, 0)\).
- Positive x-values lie to the right, while negative x-values lie to the left.
- Positive y-values are above the origin, whereas negative y-values are below.
Understanding how to navigate the coordinate plane is essential for plotting points and graphing equations accurately.
Plotting Points and Drawing Lines
Plotting points on a coordinate plane involves placing the respective x and y values on their axes. This is crucial for graphing equations accurately.
Follow these steps for plot and sketch lines:
This line represents the solution to the linear equation. Plotting points and drawing them help bring algebraic equations to visual life, showcasing the relationship between variables.
Follow these steps for plot and sketch lines:
- Identify key points, such as intercepts found through solving the equation.
- Mark these points on the coordinate plane, using the coordinates (for instance, \((3, 0)\) and \((0, -2)\)).
- Connect these points with a straight line, extending in both directions.
This line represents the solution to the linear equation. Plotting points and drawing them help bring algebraic equations to visual life, showcasing the relationship between variables.
Other exercises in this chapter
Problem 68
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