Question

a line passes through the point \\((-2, -5)\\) and has a slope of \\(-6\\). write an equation for this line.

Explanation:

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<reasoning>

Identify the given parameters

We are given a point through which the line passes and the slope of the line:

• Given point: \((x_1, y_1) = (-2, -5)\)
• Given slope: \(m = -6\)

Apply the point-slope formula

Using the Linear Equations knowledge point, we set up the point-slope equation of a line:
\[
y - y_1 = m(x - x_1)
\]
Substituting the given values:
\[
y - (-5) = -6(x - (-2))
\]
Simplifying the signs:
\[
y + 5 = -6(x + 2)
\]

Convert to slope-intercept form

Using the Slope-Intercept Form knowledge point
\[

$$\begin{aligned} y + 5 &= -6x - 12\\ y &= -6x - 17 \end{aligned}$$

\]
</reasoning>

<answer>
\(y = -6x - 17\)
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[-2, -5]],
"properties": {
"name": "(-2, -5)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "-6*x - 17", "latex": "y = -6x - 17"}, -5, 1],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2
}
}
]
}
</plot>

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Answer:

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<reasoning>

Identify the given parameters

We are given a point through which the line passes and the slope of the line:

• Given point: \((x_1, y_1) = (-2, -5)\)
• Given slope: \(m = -6\)

Apply the point-slope formula

Using the Linear Equations knowledge point, we set up the point-slope equation of a line:
\[
y - y_1 = m(x - x_1)
\]
Substituting the given values:
\[
y - (-5) = -6(x - (-2))
\]
Simplifying the signs:
\[
y + 5 = -6(x + 2)
\]

Convert to slope-intercept form

Using the Slope-Intercept Form knowledge point
\[

$$\begin{aligned} y + 5 &= -6x - 12\\ y &= -6x - 17 \end{aligned}$$

\]
</reasoning>

<answer>
\(y = -6x - 17\)
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[-2, -5]],
"properties": {
"name": "(-2, -5)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "-6*x - 17", "latex": "y = -6x - 17"}, -5, 1],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2
}
}
]
}
</plot>

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