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Sagot :
To graph the linear equation [tex]\(3x + 5y = -15\)[/tex], we will follow a step-by-step approach:
### Step 1: Rearrange the Equation in Slope-Intercept Form
First, we need to rewrite the equation in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope and [tex]\(b\)[/tex] represents the y-intercept.
Given:
[tex]\[3x + 5y = -15\][/tex]
To isolate [tex]\(y\)[/tex] on one side, we follow these steps:
1. Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[5y = -3x - 15\][/tex]
2. Divide each term by 5:
[tex]\[y = -\frac{3}{5}x - 3\][/tex]
Now, we have it in the form [tex]\(y = mx + b\)[/tex]:
[tex]\[ y = -\frac{3}{5}x - 3 \][/tex]
### Step 2: Identify Slope and Y-Intercept
From the equation [tex]\(y = -\frac{3}{5}x - 3\)[/tex], we can identify:
- Slope ([tex]\(m\)[/tex]): [tex]\(-\frac{3}{5}\)[/tex]
- Y-intercept ([tex]\(b\)[/tex]): [tex]\(-3\)[/tex]
### Step 3: Plot the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. Here, the y-intercept is [tex]\((0, -3)\)[/tex]. So, plot the point [tex]\((0, -3)\)[/tex] on the graph.
### Step 4: Use the Slope to Find Another Point
The slope [tex]\(-\frac{3}{5}\)[/tex] indicates that for every increase of 5 units in [tex]\(x\)[/tex], [tex]\(y\)[/tex] decreases by 3 units. Starting from the y-intercept [tex]\((0, -3)\)[/tex]:
1. Move 5 units to the right (positive direction) in the [tex]\(x\)[/tex]-axis: [tex]\((0 + 5, -3)\)[/tex]
2. From there, move 3 units down (in the negative direction) in the [tex]\(y\)[/tex]-axis: [tex]\((5, -3 - 3) = (5, -6)\)[/tex]
Plot the point [tex]\((5, -6)\)[/tex] on the graph.
### Step 5: Draw the Line
With the points [tex]\((0, -3)\)[/tex] and [tex]\((5, -6)\)[/tex] marked, draw a straight line passing through these points. This is the graph of the equation [tex]\(3x + 5y = -15\)[/tex].
### Step 6: Verify Additional Points (Optional)
You can pick additional [tex]\(x\)[/tex] values and compute corresponding [tex]\(y\)[/tex] values to ensure the line is plotted correctly. For example:
1. Let [tex]\(x = -5\)[/tex]:
[tex]\[3(-5) + 5y = -15 \][/tex]
[tex]\[-15 + 5y = -15 \][/tex]
[tex]\[5y = 0 \][/tex]
[tex]\[y = 0 \][/tex]
The point is [tex]\((-5, 0)\)[/tex], which should lie on the line.
### Final Graph
- Y-intercept: [tex]\((0, -3)\)[/tex]
- Second point: [tex]\((5, -6)\)[/tex]
- Equation: [tex]\(y = -\frac{3}{5}x - 3\)[/tex]
With these two points and optional verification, you can draw a straight line representing the graph of [tex]\(3x + 5y = -15\)[/tex]. The line should cross the y-axis at [tex]\((0, -3)\)[/tex] and the x-axis at the point where [tex]\(y = 0\)[/tex].
By following these steps, you will have accurately plotted the graph for the given linear equation.
### Step 1: Rearrange the Equation in Slope-Intercept Form
First, we need to rewrite the equation in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope and [tex]\(b\)[/tex] represents the y-intercept.
Given:
[tex]\[3x + 5y = -15\][/tex]
To isolate [tex]\(y\)[/tex] on one side, we follow these steps:
1. Subtract [tex]\(3x\)[/tex] from both sides:
[tex]\[5y = -3x - 15\][/tex]
2. Divide each term by 5:
[tex]\[y = -\frac{3}{5}x - 3\][/tex]
Now, we have it in the form [tex]\(y = mx + b\)[/tex]:
[tex]\[ y = -\frac{3}{5}x - 3 \][/tex]
### Step 2: Identify Slope and Y-Intercept
From the equation [tex]\(y = -\frac{3}{5}x - 3\)[/tex], we can identify:
- Slope ([tex]\(m\)[/tex]): [tex]\(-\frac{3}{5}\)[/tex]
- Y-intercept ([tex]\(b\)[/tex]): [tex]\(-3\)[/tex]
### Step 3: Plot the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. Here, the y-intercept is [tex]\((0, -3)\)[/tex]. So, plot the point [tex]\((0, -3)\)[/tex] on the graph.
### Step 4: Use the Slope to Find Another Point
The slope [tex]\(-\frac{3}{5}\)[/tex] indicates that for every increase of 5 units in [tex]\(x\)[/tex], [tex]\(y\)[/tex] decreases by 3 units. Starting from the y-intercept [tex]\((0, -3)\)[/tex]:
1. Move 5 units to the right (positive direction) in the [tex]\(x\)[/tex]-axis: [tex]\((0 + 5, -3)\)[/tex]
2. From there, move 3 units down (in the negative direction) in the [tex]\(y\)[/tex]-axis: [tex]\((5, -3 - 3) = (5, -6)\)[/tex]
Plot the point [tex]\((5, -6)\)[/tex] on the graph.
### Step 5: Draw the Line
With the points [tex]\((0, -3)\)[/tex] and [tex]\((5, -6)\)[/tex] marked, draw a straight line passing through these points. This is the graph of the equation [tex]\(3x + 5y = -15\)[/tex].
### Step 6: Verify Additional Points (Optional)
You can pick additional [tex]\(x\)[/tex] values and compute corresponding [tex]\(y\)[/tex] values to ensure the line is plotted correctly. For example:
1. Let [tex]\(x = -5\)[/tex]:
[tex]\[3(-5) + 5y = -15 \][/tex]
[tex]\[-15 + 5y = -15 \][/tex]
[tex]\[5y = 0 \][/tex]
[tex]\[y = 0 \][/tex]
The point is [tex]\((-5, 0)\)[/tex], which should lie on the line.
### Final Graph
- Y-intercept: [tex]\((0, -3)\)[/tex]
- Second point: [tex]\((5, -6)\)[/tex]
- Equation: [tex]\(y = -\frac{3}{5}x - 3\)[/tex]
With these two points and optional verification, you can draw a straight line representing the graph of [tex]\(3x + 5y = -15\)[/tex]. The line should cross the y-axis at [tex]\((0, -3)\)[/tex] and the x-axis at the point where [tex]\(y = 0\)[/tex].
By following these steps, you will have accurately plotted the graph for the given linear equation.
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