Avoiding errors calculating y-intercept with slope and a given point.

📚 Understanding the Y-Intercept

The y-intercept is the point where a line crosses the y-axis. It's the value of $y$ when $x = 0$. Knowing the slope ($m$) and a point $(x_1, y_1)$ on the line, we can find the y-intercept ($b$) using the slope-intercept form of a linear equation: $y = mx + b$. Avoiding common errors involves careful substitution and algebraic manipulation.

📜 Historical Context

The concept of coordinate geometry, which includes the slope-intercept form, was largely developed by René Descartes in the 17th century. His work provided a way to link algebra and geometry, making it possible to represent lines and curves with equations. Understanding the y-intercept is a fundamental skill that has been used for centuries in various fields.

🔑 Key Principles to Avoid Errors

• ✔️ Correct Substitution: Ensure you correctly substitute the values of $x_1$, $y_1$, and $m$ into the equation $y_1 = mx_1 + b$. Double-check the signs!
• 🧮 Algebraic Manipulation: Isolate $b$ correctly. This usually involves subtracting $mx_1$ from both sides of the equation: $b = y_1 - mx_1$.
• ➕ Sign Awareness: Pay close attention to the signs of $m$, $x_1$, and $y_1$. A negative slope or coordinate can easily lead to errors if not handled carefully.
• 🧐 Order of Operations: Follow the correct order of operations (PEMDAS/BODMAS) when calculating $b$. Multiply $m$ and $x_1$ before subtracting the result from $y_1$.
• 📝 Double-Checking: After finding $b$, plug it back into the slope-intercept form along with your given point to verify that the equation holds true.

📈 Real-World Examples

Example 1:

Find the y-intercept of a line with a slope of $m = 2$ that passes through the point $(3, 5)$.

1. Substitute the values into the equation: $5 = 2(3) + b$
2. Simplify: $5 = 6 + b$
3. Solve for $b$: $b = 5 - 6 = -1$
4. The y-intercept is $-1$.

Example 2:

Find the y-intercept of a line with a slope of $m = -3$ that passes through the point $(-1, 4)$.

1. Substitute the values into the equation: $4 = -3(-1) + b$
2. Simplify: $4 = 3 + b$
3. Solve for $b$: $b = 4 - 3 = 1$
4. The y-intercept is $1$.

Example 3:

Find the y-intercept of a line with a slope of $m = \frac{1}{2}$ that passes through the point $(4, -2)$.

1. Substitute the values into the equation: $-2 = \frac{1}{2}(4) + b$
2. Simplify: $-2 = 2 + b$
3. Solve for $b$: $b = -2 - 2 = -4$
4. The y-intercept is $-4$.

📝 Practice Quiz

Find the y-intercept for each of the following:

1. A line with slope $m = 3$ passing through $(2, 8)$.
2. A line with slope $m = -2$ passing through $(-1, 5)$.
3. A line with slope $m = \frac{1}{3}$ passing through $(6, 1)$.
4. A line with slope $m = -\frac{1}{2}$ passing through $(4, -3)$.
5. A line with slope $m = 0.5$ passing through $(2, 4)$.
6. A line with slope $m = -1.5$ passing through $(-2, 1)$.
7. A line with slope $m = \frac{2}{3}$ passing through $(3, -2)$.

💡 Conclusion

Finding the y-intercept given the slope and a point is a fundamental skill in algebra. By carefully substituting values, paying attention to signs, and correctly manipulating the equation, you can avoid common errors and master this concept. Practice regularly and double-check your work to build confidence and accuracy. Good luck! 👍