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Find the y-intercept of the line with the equation \( y = -\frac{1}{2}x + 5 \).

Ask by Nguyen Cook. in Nigeria
Feb 03,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

The y-intercept is 5, which is the point (0, 5).

Solution

To find the **y-intercept** of the line given by the equation: \[ y = -\frac{1}{2}x + 5 \] **Step 1: Understand the Equation** The equation is in the slope-intercept form: \[ y = mx + b \] where: - \( m \) is the **slope** of the line. - \( b \) is the **y-intercept** of the line. **Step 2: Identify the Y-Intercept** From the given equation: \[ y = -\frac{1}{2}x + 5 \] - The slope (\( m \)) is \( -\frac{1}{2} \). - The y-intercept (\( b \)) is **5**. **Step 3: Express the Y-Intercept as a Point** The y-intercept is the point where the line crosses the y-axis. At this point, the value of \( x \) is **0**. Therefore, the y-intercept can be expressed as the point: \[ (0, 5) \] **Conclusion** The y-intercept of the line \( y = -\frac{1}{2}x + 5 \) is **5**, which corresponds to the point **(0, 5)** on the coordinate plane.

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The Deep Dive

To find the y-intercept of the line described by the equation \( y = -\frac{1}{2}x + 5 \), you simply need to determine the value of \( y \) when \( x = 0 \). Plugging in \( 0 \) for \( x \), the equation becomes \( y = -\frac{1}{2}(0) + 5 \), which simplifies to \( y = 5 \). Therefore, the y-intercept is the point \( (0, 5) \). Did you know that the y-intercept is a crucial point in graphing lines? It's the point where the line crosses the y-axis, serving as a key reference point. This makes it super helpful when sketching graphs quickly! Moreover, finding intercepts like this is often very useful in real-world applications, such as determining where a budget meets the income line in finance, or forecasting where demand meets supply in economics. It helps in understanding various linear relationships in everyday life!

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