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Chapter 10: Problem 9

In Exercises 1 to \(8,\) draw the graph of each equation. Name any intencepts. Which point(s) lie on the graph of \(5 x-4 y=32 ?\) a) \(\quad(4,-3)\) b) \((8,1)\)

Short Answer

Expert verified
The point \((4, -3)\) lies on the graph, and \((8, 1)\) does not.

Step by step solution

01

Plug Point (4,-3) Into Equation

Start by substituting the point \((4, -3)\) into the equation. Substitute \(x = 4\) and \(y = -3\) into the equation \(5x - 4y = 32\). This gives us:\[5(4) - 4(-3) = 32\] \[20 + 12 = 32\] Since \(32 = 32\), the point \((4, -3)\) lies on the graph.
02

Plug Point (8,1) Into Equation

Now, substitute the point \((8, 1)\) into the equation. Substitute \(x = 8\) and \(y = 1\) into the equation \(5x - 4y = 32\). This gives us:\[5(8) - 4(1) = 32\] \[40 - 4 = 36\] Since \(36 eq 32\), the point \((8, 1)\) does not lie on the graph.

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Key Concepts

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

Intercepts
Intercepts are crucial concepts in graphing linear equations and understanding coordinate geometry. Intercepts are the points where a graph crosses the axes. There are two types:
  • x-intercept: The point where the graph crosses the x-axis. Here, the y-value is zero.
  • y-intercept: The point where the graph crosses the y-axis. Here, the x-value is zero.
To find these intercepts from an equation, you can use substitution:
  • For the x-intercept, set y = 0 in the equation and solve for x.
  • For the y-intercept, set x = 0 in the equation and solve for y.
Understanding these intersection points helps provide a sense of the direction and location of the line on a graph, allowing you to sketch it effectively. Identifying intercepts is a fundamental step in graphing linear equations because it helps to check the consistency of your line with plotted data.
Coordinate Geometry
Coordinate geometry, or analytic geometry, is a branch of mathematics that uses coordinates to describe geometric figures and analyze their properties. In coordinate geometry, every point is represented by a pair of numbers \(x, y\), indicating its position on a plane defined by an x-axis (horizontal) and a y-axis (vertical). Graphing linear equations is a primary application of coordinate geometry.To graph the equation \(5x - 4y = 32\), you would typically:
  • Find two or more points that satisfy the equation using equation substitution.
  • Plot these points on the \(x, y\) coordinate plane.
  • Draw a straight line through these points to represent the equation.
The coordinate plane is divided into four quadrants, allowing us to locate points based on the sign and value of their coordinates, helping in visualizing how the linear equations behave along these axes.
Equation Substitution
Equation substitution is a method used to verify if a point lies on the graph of a given linear equation. This process requires substituting the x and y coordinates from a point into the equation to check if the equation holds true (balances).For example, consider the equation \(5x - 4y = 32\):
  • Substitute x = 4 and y = -3 into the equation to check if (4, -3) is on the line:
    \[5(4) - 4(-3) = 32\] results in \[20 + 12 = 32\]
  • If both sides of the equation are equal, the point lies on the graph, as it does here because \[32 = 32\].
  • Substitute x = 8 and y = 1 into the equation to check the point (8, 1):
    \[5(8) - 4(1) = 32\] results in \[40 - 4 = 36\]
  • Since \[36 eq 32\], this point does not lie on the graph.
This method simplifies the verification process by working directly with the equation, making it a powerful tool in graphing and understanding the relationships between equations and their graphical representations.

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