Problem 19 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Through \((2,1)\) and \((1,6)\)

Step-by-Step Solution

Answer

The equation of the line is \( y = -5x + 11 \).

Step 1: Calculate Slope

To find the slope (m) of the line, we use the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Plugging in the given points \((x_1, y_1) = (2, 1)\) and \((x_2, y_2) = (1, 6)\), we get: \( m = \frac{6 - 1}{1 - 2} = \frac{5}{-1} = -5 \). Thus, the slope is \( -5 \).

Step 2: Use Point-Slope Form

The point-slope formula of a line is \( y - y_1 = m(x - x_1) \). Using the slope \( m = -5 \) and the point \((2, 1)\), substitute into the formula: \( y - 1 = -5(x - 2) \).

Step 3: Simplify to Slope-Intercept Form

To convert the equation to slope-intercept form \( y = mx + b \), distribute and simplify: \( y - 1 = -5x + 10 \). Adding 1 to both sides gives \( y = -5x + 11 \).

Step 4: Verify with Second Point

Ensure the equation passes through the second point \((1, 6)\). Substitute \( x = 1 \) into the equation: \( y = -5(1) + 11 = -5 + 11 = 6 \). This confirms the equation works.

Key Concepts

Slope

The slope of a line is a measure that indicates how steep the line is. You calculate the slope by dividing the change in the y-values by the change in the x-values between two points on the line. This is often described as "rise over run."
To find the slope of a line given two points, you can use the formula \[m = \frac{y_2 - y_1}{x_2 - x_1}.\]
You just substitute the coordinates of the points into this formula.

• In our exercise, we calculated the slope between the points \((2, 1)\) and \((1, 6)\).
• Using the formula, the calculation is \(-5 = \frac{6 - 1}{1 - 2}\).
• A negative result means the line decreases from left to right, whereas a positive slope indicates an upward trend.

Understanding the slope is important because it tells us how the line moves across the coordinate plane. A zero slope indicates a horizontal line, while an undefined slope means the line is vertical.

Point-Slope Form

The point-slope form of a line’s equation is useful because it allows you to write the equation of a line if you know one point on the line and its slope. The form is \[y - y_1 = m(x - x_1)\],where \(m\)is the slope, and \(x_1, y_1\) is a point on the line.
In the given exercise:

• We used the slope \(m = -5\)and the point \((2, 1)\).
• Substituting these into the equation, we obtain \(y - 1 = -5(x - 2).\)

The point-slope form can be directly converted into other forms like the slope-intercept form. This versatility helps when working with different types of problems in algebra and geometry. It's crucial when you need to quickly find the line's equation based on minimal information.

Slope-Intercept Form

The slope-intercept form of a line's equation is a way to express the line in an easily interpretable form. This equation format is \[y = mx + b\],where \(m\) is the slope and \(b\) is the y-intercept, the point where the line crosses the y-axis. This form is practical because it immediately reveals the slope and y-intercept.
To convert the point-slope form into the slope-intercept form in our exercise:

• We started with \(y - 1 = -5(x - 2)\),expanded it to \(y = -5x + 11\).

The slope-intercept form makes graphing the line straightforward as you can easily locate the y-intercept and plot the line’s steepness using the slope. This form is often preferred for graphing and quickly understanding the relationship between variables in the equation.