Question
103 Find an equabon for the Ine with the given properics. Express your answer using ether the general form on the slope-in:ercept form of the equa Slope \( =2 \) corlaining the point \( (-3,-1) \)
Ask by Perkins Vargas. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The equation of the line is \( y = 2x + 5 \) or \( 2x - y + 5 = 0 \).
Solution
1. Start with the slope-intercept form of a line:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the \( y \)-intercept.
2. Substitute the given slope \( m = 2 \) into the equation:
\[
y = 2x + b
\]
3. Use the given point \((-3, -1)\) to find \( b \). Substitute \( x = -3 \) and \( y = -1 \):
\[
-1 = 2(-3) + b
\]
4. Simplify the equation:
\[
-1 = -6 + b
\]
5. Solve for \( b \) by adding \( 6 \) to both sides:
\[
b = -1 + 6 = 5
\]
6. Write the equation in slope-intercept form:
\[
y = 2x + 5
\]
7. Alternatively, convert the equation to general form. Start with:
\[
y = 2x + 5
\]
Rearrange the terms to bring all terms to one side:
\[
2x - y + 5 = 0
\]
Thus, the equation of the line can be expressed as either:
- Slope-intercept form: \( y = 2x + 5 \)
- General form: \( 2x - y + 5 = 0 \)
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Simplify this solution Mind Expander
To find the equation of a line with a slope of \(2\) that passes through the point \((-3, -1)\), you can use the point-slope form of a linear equation, which is \(y - y_1 = m(x - x_1)\). Plugging in the values, you get \(y - (-1) = 2(x - (-3))\). Simplifying this results in \(y + 1 = 2(x + 3)\), which can be expanded to \(y + 1 = 2x + 6\), and finally, \(y = 2x + 5\). So, the slope-intercept form of the equation is \(y = 2x + 5\). If you're looking to convert this into the general form \(Ax + By + C = 0\), you can rearrange it to \(2x - y + 5 = 0\). Either form is correct; it just depends on what you prefer for your equation!
