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The equation of the line is \( y = -2x + 6 \).
Solución
To find the equation of a line given a point and a slope, we can use the point-slope form of the equation of a line, which is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is the point on the line and \( m \) is the slope.
### Given:
- Point: \( (1, 4) \)
- Slope: \( m = -2 \) (using the negative slope)
### Step 1: Substitute the values into the point-slope form
Substituting \( x_1 = 1 \), \( y_1 = 4 \), and \( m = -2 \):
\[
y - 4 = -2(x - 1)
\]
### Step 2: Simplify the equation
Now, we will simplify the equation:
\[
y - 4 = -2x + 2
\]
Adding 4 to both sides:
\[
y = -2x + 6
\]
### Final Equation
The equation of the line in slope-intercept form is:
\[
y = -2x + 6
\]
This represents the line that passes through the point \( (1, 4) \) with a slope of \( -2 \).
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Simplifique esta solución Beyond the Answer
To find the equation of a line given a point and a slope, you can utilize the point-slope form of a linear equation, which is \(y - y_1 = m(x - x_1)\). Here, you have the point \((1, 4)\) and the slope \(-2\) (using the negative slope). Plugging these values in, the equation becomes \(y - 4 = -2(x - 1)\). Simplifying it leads to \(y = -2x + 6\) in slope-intercept form. Now, if you were to use the positive slope of \(2\), the equation would be \(y - 4 = 2(x - 1)\), leading to \(y = 2x + 2\). So, you can create two lines from this point: one sloping downward and one sloping upward! Such versatility shows how lines can represent various relationships in different contexts!
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