Join the conversation on IDNLearn.com and get the answers you seek from experts. Get step-by-step guidance for all your technical questions from our knowledgeable community members.
Sagot :
To graph the line [tex]$y = 1 - \frac{7}{4}(x - 4)$[/tex], we will follow a step-by-step approach:
### 1. Simplify the equation
First, let's simplify the given equation:
[tex]\[ y = 1 - \frac{7}{4}(x - 4) \][/tex]
Distribute [tex]\( \frac{7}{4} \)[/tex]:
[tex]\[ y = 1 - \frac{7}{4}x + \frac{7}{4} \times 4 \][/tex]
[tex]\[ y = 1 - \frac{7}{4}x + 7 \][/tex]
Combine the constant terms:
[tex]\[ y = -\frac{7}{4}x + 8 \][/tex]
Now, the equation is in the slope-intercept form [tex]\( y = mx + b \)[/tex] where the slope [tex]\( m = -\frac{7}{4} \)[/tex] and the y-intercept [tex]\( b = 8 \)[/tex].
### 2. Plot the y-intercept
The y-intercept [tex]\( b \)[/tex] is the point where the line crosses the y-axis. It occurs when [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 8 \][/tex]
So, the y-intercept is the point [tex]\( (0, 8) \)[/tex]. Plot this point on the graph.
### 3. Use the slope to find another point
The slope [tex]\( m \)[/tex] of the line is [tex]\( -\frac{7}{4} \)[/tex]. This means that for every 4 units you move to the right, you move 7 units down.
Starting from the y-intercept [tex]\( (0, 8) \)[/tex]:
- Move 4 units to the right (increasing [tex]\( x \)[/tex] by 4): [tex]\( x = 4 \)[/tex]
- Then move 7 units down (decreasing [tex]\( y \)[/tex] by 7): [tex]\( y = 8 - 7 = 1 \)[/tex]
So, another point on the line is [tex]\( (4, 1) \)[/tex]. Plot this point on the graph.
### 4. Draw the line
Once you have two points [tex]\( (0, 8) \)[/tex] and [tex]\( (4, 1) \)[/tex], you can draw a straight line through these points. This line represents the graph of the equation [tex]\( y = -\frac{7}{4}x + 8 \)[/tex].
### 5. Verify by choosing another point
For verification, you can choose another [tex]\( x \)[/tex]-value, substitute it into the original equation, and see if it lies on the drawn line.
For instance, let’s choose [tex]\( x = -4 \)[/tex]:
[tex]\[ y = 1 - \frac{7}{4}(-4 - 4) \][/tex]
[tex]\[ y = 1 - \frac{7}{4}(-8) \][/tex]
[tex]\[ y = 1 + 14 \][/tex]
[tex]\[ y = 15 \][/tex]
So, the point [tex]\( (-4, 15) \)[/tex] should also lie on the line. Plotting this point can serve as confirmation that your line is correct.
### Summary Diagram
[tex]\[ \begin{align*} \text{Points:} & \ (0, 8), (4, 1), (-4, 15) \end{align*} \][/tex]
### Graph
Below is a conceptual representation of how the graph should appear:
[tex]\[ \begin{tikzpicture} \begin{axis}[ axis lines = middle, xlabel = \(x\), ylabel = \(y\), grid = both ] % Lines \addplot[domain=-10:10, samples=200, color=blue]{-1.75 * x + 8}; % Points \addplot[only marks, color=red] coordinates {(0,8) (4,1) (-4,15)}; \end{axis} \end{tikzpicture} \][/tex]
### Final Thoughts
- The line has a negative slope, so it should slant downwards from left to right.
- The plotted points [tex]\( (0, 8) \)[/tex], [tex]\( (4, 1) \)[/tex], and [tex]\( (-4, 15) \)[/tex] should lie on the straight line belonging to the equation [tex]\( y = 1 - \frac{7}{4}(x - 4) \)[/tex].
With this approach, you should be able to accurately sketch the graph for the given linear equation.
### 1. Simplify the equation
First, let's simplify the given equation:
[tex]\[ y = 1 - \frac{7}{4}(x - 4) \][/tex]
Distribute [tex]\( \frac{7}{4} \)[/tex]:
[tex]\[ y = 1 - \frac{7}{4}x + \frac{7}{4} \times 4 \][/tex]
[tex]\[ y = 1 - \frac{7}{4}x + 7 \][/tex]
Combine the constant terms:
[tex]\[ y = -\frac{7}{4}x + 8 \][/tex]
Now, the equation is in the slope-intercept form [tex]\( y = mx + b \)[/tex] where the slope [tex]\( m = -\frac{7}{4} \)[/tex] and the y-intercept [tex]\( b = 8 \)[/tex].
### 2. Plot the y-intercept
The y-intercept [tex]\( b \)[/tex] is the point where the line crosses the y-axis. It occurs when [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 8 \][/tex]
So, the y-intercept is the point [tex]\( (0, 8) \)[/tex]. Plot this point on the graph.
### 3. Use the slope to find another point
The slope [tex]\( m \)[/tex] of the line is [tex]\( -\frac{7}{4} \)[/tex]. This means that for every 4 units you move to the right, you move 7 units down.
Starting from the y-intercept [tex]\( (0, 8) \)[/tex]:
- Move 4 units to the right (increasing [tex]\( x \)[/tex] by 4): [tex]\( x = 4 \)[/tex]
- Then move 7 units down (decreasing [tex]\( y \)[/tex] by 7): [tex]\( y = 8 - 7 = 1 \)[/tex]
So, another point on the line is [tex]\( (4, 1) \)[/tex]. Plot this point on the graph.
### 4. Draw the line
Once you have two points [tex]\( (0, 8) \)[/tex] and [tex]\( (4, 1) \)[/tex], you can draw a straight line through these points. This line represents the graph of the equation [tex]\( y = -\frac{7}{4}x + 8 \)[/tex].
### 5. Verify by choosing another point
For verification, you can choose another [tex]\( x \)[/tex]-value, substitute it into the original equation, and see if it lies on the drawn line.
For instance, let’s choose [tex]\( x = -4 \)[/tex]:
[tex]\[ y = 1 - \frac{7}{4}(-4 - 4) \][/tex]
[tex]\[ y = 1 - \frac{7}{4}(-8) \][/tex]
[tex]\[ y = 1 + 14 \][/tex]
[tex]\[ y = 15 \][/tex]
So, the point [tex]\( (-4, 15) \)[/tex] should also lie on the line. Plotting this point can serve as confirmation that your line is correct.
### Summary Diagram
[tex]\[ \begin{align*} \text{Points:} & \ (0, 8), (4, 1), (-4, 15) \end{align*} \][/tex]
### Graph
Below is a conceptual representation of how the graph should appear:
[tex]\[ \begin{tikzpicture} \begin{axis}[ axis lines = middle, xlabel = \(x\), ylabel = \(y\), grid = both ] % Lines \addplot[domain=-10:10, samples=200, color=blue]{-1.75 * x + 8}; % Points \addplot[only marks, color=red] coordinates {(0,8) (4,1) (-4,15)}; \end{axis} \end{tikzpicture} \][/tex]
### Final Thoughts
- The line has a negative slope, so it should slant downwards from left to right.
- The plotted points [tex]\( (0, 8) \)[/tex], [tex]\( (4, 1) \)[/tex], and [tex]\( (-4, 15) \)[/tex] should lie on the straight line belonging to the equation [tex]\( y = 1 - \frac{7}{4}(x - 4) \)[/tex].
With this approach, you should be able to accurately sketch the graph for the given linear equation.
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and see you next time for more reliable information.
