Problem 54 - Algebra 2

Question

CITY PLANNING. City planners have laid out streets on a coordinate grid before beginning construction. One street lies on the line with equation \(y=2 x+1 .\) Another street that intersects the first street passes through the point \((2,-3)\) and is perpendicular to the first street. What is the equation of the line on which the second street lies?

Step-by-Step Solution

Answer

Equation is \(y = -\frac{1}{2}x - 2\)

Step 1: Understand the Slope of the First Street

The equation of the first street is given as \(y = 2x + 1\). This is in slope-intercept form \(y = mx + b\), where \(m\) is the slope. Therefore, the slope of the first street is 2.

Step 2: Determine the Slope of the Perpendicular Line

Lines that are perpendicular have slopes that are negative reciprocals of each other. If the slope of the first street (m) is 2, then the slope (m') of the perpendicular line is \(-\frac{1}{2}\) because \(-\frac{1}{2}\) is the negative reciprocal of 2.

Step 3: Use Point-Slope Formula

The equation of a line can be found using the point-slope formula: \(y - y_1 = m(x - x_1)\). Substitute the slope \(m = -\frac{1}{2}\) and the given point \((2,-3)\) into the formula. This gives: \[y + 3 = -\frac{1}{2}(x - 2)\]

Step 4: Simplify to Slope-Intercept Form

Expand and simplify the equation: \[y + 3 = -\frac{1}{2}x + 1\]. To isolate \(y\), subtract 3 from both sides, giving \[y = -\frac{1}{2}x + 1 - 3\] which simplifies to \[y = -\frac{1}{2}x - 2\].

Key Concepts

Perpendicular Lines

Understanding perpendicular lines is crucial when working with coordinate geometry and city planning. Perpendicular lines intersect at a 90-degree angle. This unique property makes them particularly helpful in creating well-organized and navigable street grids.

In mathematical terms, when two lines are perpendicular, the product of their slopes is \(-1\). If you have the slope of one line, say \(m\), the slope of the perpendicular line will be the negative reciprocal, which is \(-\frac{1}{m}\). For example, if the slope of one line is 2, the slope of its perpendicular line will be \(-\frac{1}{2}\). This concept is a staple in coordinate geometry and invaluable for designing systems like city streets that need to meet at right angles.

Slope-Intercept Form

The slope-intercept form of a linear equation is perhaps the most intuitive way to describe a line on a coordinate plane. The equation is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This form is called "slope-intercept" because it visibly shows you the slope and the point where the line crosses the y-axis.

Here's why it's so useful:

• The slope \(m\) tells you how steep the line is. A higher slope means a steeper line.
• The intercept \(b\) tells you where the line will cross the y-axis, making it handy for sketching graphs.

With the slope and y-intercept outlined clearly, you can easily graph the line by starting at the intercept (the point \((0, b)\)) and using the slope to find other points.

Point-Slope Formula

When you know a point on a line and the slope, the point-slope formula is your go-to tool. This formula is written as \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a specific point on the line.

The point-slope formula is particularly handy when the line's equation needs to include a specific point. Here's how you use it:

• Identify your slope \(m\) and the coordinates of your point \((x_1, y_1)\).
• Substitute these values into the formula to find the general equation of your line.
• From here, you can even convert it to slope-intercept form by solving for \(y\) to get \(y = mx + b\).

This method is especially useful in scenarios like city planning, where connecting specific pre-determined points with straight paths (or roads) is necessary.