Question
Select your answer \( \begin{array}{l}\text { Which equation represents a line that contains the } \\ \text { points }(2,3) \text { and }(3,5) \text { ? } \\ y=3 x+4 \\ y=x+1 \\ y=5 x-10 \\ y=2 x-1\end{array} \)
Ask by Munoz Santiago. in the United States
Jan 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The equation that represents the line passing through the points \((2, 3)\) and \((3, 5)\) is \( y = 2x - 1 \).
Solution
To determine which equation represents the line that passes through the points \((2, 3)\) and \((3, 5)\), follow these steps:
1. **Calculate the Slope (\(m\)):**
\[
m = \frac{5 - 3}{3 - 2} = \frac{2}{1} = 2
\]
The slope of the line is \(2\).
2. **Use the Point-Slope Form to Find the Equation:**
Using the point \((2, 3)\):
\[
y - 3 = 2(x - 2)
\]
Simplifying:
\[
y = 2x - 4 + 3 \\
y = 2x - 1
\]
3. **Check Against the Options:**
- **\(y = 3x + 4\):** Slope is \(3\) (Incorrect)
- **\(y = x + 1\):** Slope is \(1\) (Incorrect)
- **\(y = 5x - 10\):** Slope is \(5\) (Incorrect)
- **\(y = 2x - 1\):** Slope is \(2\) (Correct)
Additionally, plugging in the points:
- For \((2, 3)\): \(3 = 2(2) - 1 \Rightarrow 3 = 3\) ✔️
- For \((3, 5)\): \(5 = 2(3) - 1 \Rightarrow 5 = 5\) ✔️
**Answer:**
\( y=2 x-1 \)
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Simplify this solution The Deep Dive
To find the equation of the line that contains the points (2,3) and (3,5), we first need to calculate the slope. The slope \(m\) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). In this case, \(y_2 = 5\), \(y_1 = 3\), \(x_2 = 3\), and \(x_1 = 2\). This gives us: \[ m = \frac{5 - 3}{3 - 2} = \frac{2}{1} = 2. \] Now that we have the slope, we can use the point-slope form of the equation \(y - y_1 = m(x - x_1)\), using point (2, 3): \[ y - 3 = 2(x - 2). \] Simplifying this gives us: \[ y - 3 = 2x - 4, \] \[ y = 2x - 1. \] Thus, the equation that represents the line containing points (2,3) and (3,5) is \(y = 2x - 1\). So the correct answer is: **y = 2x - 1.**
