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Magazine Chapter 3: Problem 45
Given the set of \(x\) -values \(\\{-2,-1,0,1,2\\},\) find the corresponding \(y\) -values and graph them. $$ 3 x-y=9 $$
Short Answer
Expert verified
The y-values are -15, -12, -9, -6, -3 for x-values -2, -1, 0, 1, 2, respectively.
Step by step solution
01
Understand the equation
We start with the equation \(3x - y = 9\). The goal is to express \(y\) in terms of \(x\) so that we can easily substitute the given \(x\)-values to find corresponding \(y\)-values.
02
Rearrange the equation for y
Rearrange the equation \(3x - y = 9\) by solving for \(y\). Add \(y\) to both sides and subtract \(9\) to get \(y = 3x - 9\).
03
Substitute x = -2
Substitute \(x = -2\) into the equation \(y = 3x - 9\):\[y = 3(-2) - 9 = -6 - 9 = -15\]So, when \(x = -2\), \(y = -15\).
04
Substitute x = -1
Substitute \(x = -1\) into the equation \(y = 3x - 9\):\[y = 3(-1) - 9 = -3 - 9 = -12\]So, when \(x = -1\), \(y = -12\).
05
Substitute x = 0
Substitute \(x = 0\) into the equation \(y = 3x - 9\):\[y = 3(0) - 9 = 0 - 9 = -9\]So, when \(x = 0\), \(y = -9\).
06
Substitute x = 1
Substitute \(x = 1\) into the equation \(y = 3x - 9\):\[y = 3(1) - 9 = 3 - 9 = -6\]So, when \(x = 1\), \(y = -6\).
07
Substitute x = 2
Substitute \(x = 2\) into the equation \(y = 3x - 9\):\[y = 3(2) - 9 = 6 - 9 = -3\]So, when \(x = 2\), \(y = -3\).
08
Plot the points on a graph
Now that we have identified the pairs \((-2, -15), (-1, -12), (0, -9), (1, -6), (2, -3)\), plot these points on a coordinate plane. Connect the dots to see the line represented by the equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solving for Y
To solve for the variable 'y', we need to rearrange the equation so that 'y' is isolated on one side. We started with the equation \(3x - y = 9\). The trick here is to get all terms involving 'y' on one side of the equation and everything else on the other side. Here鈥檚 a simple way to solve for 'y':
- Add 'y' to both sides. This makes it easier to handle as it neatly brings 'y' to the positive side: \(3x = y + 9\).
- Next, subtract 9 from both sides. This final step isolates 'y': \(y = 3x - 9\).
Coordinate Plane
The coordinate plane is a two-dimensional surface where we can visually represent equations. It's made up of two axes:
Understanding the coordinate plane is essential for graphing equations. By plotting each point determined from our equation \(y = 3x - 9\), we can visualize how 'y' changes in relation to 'x'. Points like (-2, -15) or (0, -9) become intuitively understandable and valuable through this graphical representation.
- The horizontal axis, known as the x-axis.
- The vertical axis, referred to as the y-axis.
Understanding the coordinate plane is essential for graphing equations. By plotting each point determined from our equation \(y = 3x - 9\), we can visualize how 'y' changes in relation to 'x'. Points like (-2, -15) or (0, -9) become intuitively understandable and valuable through this graphical representation.
Graphing Linear Functions
Graphing a linear function, such as \(y = 3x - 9\), involves drawing a straight line through the points derived from this function. Linear functions are characterized by their constant rate of change, also known as slope.
To graph this equation:
To graph this equation:
- List your pairs of (x, y) values, which, for our exercise, are (-2, -15), (-1, -12), (0, -9), (1, -6), and (2, -3).
- Plot each point on the coordinate plane accurately.
- Once all points are plotted, connect them using a straight line.
