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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.
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