Problem 20 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Through \((-1,-2)\) and \((4,3)\)

Step-by-Step Solution

Answer

The equation of the line is \(y = x - 1\).

Step 1: Identify the Points

The given points are (-1, -2) and (4, 3). These points will be used to find the slope of the line and develop the equation of the line.

Step 2: Calculate the Slope

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Using the points (-1, -2) and (4, 3):\[m = \frac{3 - (-2)}{4 - (-1)} = \frac{3 + 2}{4 + 1} = \frac{5}{5} = 1\]

Step 3: Write the Point-Slope Form

The point-slope form of a line's equation is given by:\[y - y_1 = m(x - x_1)\]Substitute \(m = 1\), using point (-1, -2):\[y + 2 = 1(x + 1)\]

Step 4: Simplify to Slope-Intercept Form

Distribute and rearrange the equation from the point-slope form:\[y + 2 = x + 1\]Subtract 2 from both sides to solve for \(y\):\[y = x - 1\]

Step 5: Verify the Equation

Substitute the points back into the equation \(y = x - 1\) to ensure it satisfies both points:For (-1, -2):\[-2 = (-1) - 1 = -2\]For (4, 3):\[3 = 4 - 1 = 3\]Both points satisfy the equation.

Key Concepts

Slope Calculation

The slope of a line is a measure of its steepness. It's calculated as the ratio of the vertical change to the horizontal change between two points on the line. This is often described as "rise over run." Understanding this concept is crucial for developing an equation of the line. Given two points, \((x_1, y_1)\) and \((x_2, y_2)\), the formula for finding the slope \(m\) is:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

In our specific problem, the points \((-1, -2)\) and \(4, 3)\) are used. Calculating the slope:

• Substitute the values into the formula: \[ m = \frac{3 - (-2)}{4 - (-1)} = \frac{5}{5} = 1 \]

The slope is 1, meaning for every unit increase in x, y also increases by 1.

Point-Slope Form

Once you have the slope, you can use the point-slope formula to write an equation for the line. This form is especially useful when you know a point on the line and its slope. The formula is:

• \[ y - y_1 = m(x - x_1) \]

Here, \(m\) represents the slope calculated earlier, and \((x_1, y_1)\) is a point on the line. Using our known slope of 1 and the point \((-1, -2)\), the equation becomes:

• \[ y + 2 = 1(x + 1) \]

This simple equation reflects the line's behavior, pivoting around the point \((-1, -2)\) with the slope of 1.

Slope-Intercept Form

The slope-intercept form of a line is one of the most commonly used forms, ideal for quickly understanding a line's slope and y-intercept (where it crosses the y-axis). The formula is:

• \[ y = mx + b \]

Here, \(m\) is the slope, and \(b\) is the y-intercept. Transforming our point-slope form \((y + 2 = x + 1)\) into this form, you simplify it:

• \[ y + 2 = x + 1 \]
• Subtract 2 from both sides to isolate y: \[ y = x - 1 \]

Now, the equation \(y = x - 1\) shows both the slope (1) and the y-intercept (-1) of the line.

Verify Equation

Verifying that the equation correctly models the line through the given points is essential to ensure accuracy. This involves plugging the original points into the final equation \(y = x - 1\) and checking their validity:

• For \((-1, -2)\):
  • Substitute x with -1 and y with -2 in the equation \(y = x - 1\).
  • Result: \[-2 = (-1) - 1 = -2\]
• For \((4, 3)\):
  • Substitute x with 4 and y with 3 in the equation \(y = x - 1\).
  • Result: \[3 = 4 - 1 = 3\]

Both calculations confirm that the line passes through each point, confirming the accuracy of the line equation.