Question
7. slope \( =-\frac{3}{4} \), goes through the point \( (4,2) \)
Ask by Burgess Watson. in the United States
Mar 08,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
La ecuación de la recta es \( y = -\frac{3}{4}x + 5 \).
Solution
1. Dado que la pendiente es \( m = -\frac{3}{4} \) y la recta pasa por el punto \( (4,2) \), podemos utilizar la forma punto-pendiente:
\[
y - y_1 = m(x - x_1)
\]
2. Sustituimos \( m = -\frac{3}{4} \) y \( (x_1, y_1) = (4,2) \) en la fórmula:
\[
y - 2 = -\frac{3}{4}(x - 4)
\]
3. Expandimos el lado derecho:
\[
y - 2 = -\frac{3}{4}x + \frac{3}{4}\cdot 4 = -\frac{3}{4}x + 3
\]
4. Sumamos 2 a ambos lados para despejar la \( y \):
\[
y = -\frac{3}{4}x + 3 + 2
\]
\[
y = -\frac{3}{4}x + 5
\]
Por lo tanto, la ecuación de la recta es:
\[
y = -\frac{3}{4}x + 5
\]
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Simplify this solution Bonus Knowledge
To find the equation of a line with a slope of \( -\frac{3}{4} \) that passes through the point \( (4,2) \), you can use the point-slope form of a linear equation, which is \( y - y_1 = m(x - x_1) \). Plugging in the point and the slope, you get \( y - 2 = -\frac{3}{4}(x - 4) \). A little simplification leads to the equation \( y = -\frac{3}{4}x + 5 \). Voila! Now, if you're sketching this line on graph paper or a digital tool, start from the point \( (4,2) \). From there, remember the slope means for every 4 steps you take to the right (positive direction), you move down 3 steps (negative direction). This will help you plot your line accurately!
