Problem 13 - Algebra A Combined Function

Question

Write an equation of the line passing through the given points. Write the equation in standard form \(A x+B y=C\). See Example 2 . (-2,-4) and (-4,-3)

Step-by-Step Solution

Answer

The equation of the line is \(x + 2y = -10\).

Step 1: Find the Slope of the Line

First, calculate the slope (m) of the line passing through the given points (-2, -4) and (-4, -3) using the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the points (-2, -4) as \((x_1, y_1)\) and (-4, -3) as \((x_2, y_2)\):\[ m = \frac{-3 - (-4)}{-4 - (-2)} \]Calculate the results:\[ m = \frac{-3 + 4}{-4 + 2} = \frac{1}{-2} = -\frac{1}{2} \]

Step 2: Write the Point-Slope Form Equation

Use the point-slope form to write the equation of the line with the obtained slope and one of the points, say (-2, -4):\[ y - y_1 = m(x - x_1) \]Substitute the slope \(m = -\frac{1}{2}\) and the point (-2, -4):\[ y + 4 = -\frac{1}{2}(x + 2) \]

Step 3: Simplify and Convert to Slope-Intercept Form

Distribute the slope and simplify the point-slope equation:\[ y + 4 = -\frac{1}{2}x - 1 \]Subtract 4 from both sides to isolate \(y\):\[ y = -\frac{1}{2}x - 1 - 4 \]\[ y = -\frac{1}{2}x - 5 \]

Step 4: Convert to Standard Form

The standard form of a line equation is \(Ax + By = C\).Move \(-\frac{1}{2}x\) to the left side by adding it to both sides of the equation:\[ \frac{1}{2}x + y = -5 \]To eliminate the fraction, multiply every term by 2:\[ x + 2y = -10 \]

Step 5: Write the Final Equation

The equation \(x + 2y = -10\) is now in standard form \(Ax + By = C\), where \(A = 1\), \(B = 2\), and \(C = -10\).

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial for dealing with equations in coordinate geometry. The slope indicates how steep a line is and in which direction it inclines. It's calculated as the ratio of the change in the y-coordinates (vertical change) to the change in the x-coordinates (horizontal change) between two distinct points on a line. The formula used is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Let's break down our example with the points \((-2, -4)\) and \((-4, -3)\). By substituting into the formula, we solve:

• \( m = \frac{-3 + 4}{-4 + 2} = -\frac{1}{2} \)

Therefore, the slope is \(-\frac{1}{2}\).
This tells us the line descends to the right.

Point-Slope Form

The point-slope form is a way to write the equation of a line when you know its slope and a point on the line. It is expressed as:

• \( y - y_1 = m(x - x_1) \)

Here, \((x_1, y_1)\) is the point, and \(m\) is the slope. With our slope \(-\frac{1}{2}\) and point \((-2, -4)\), the equation becomes:

• \( y + 4 = -\frac{1}{2}(x + 2) \)

This form is particularly useful as it allows us to quickly transition to other forms of the equation, like the slope-intercept form.
Use this form to make calculations straightforward.

Slope-Intercept Form

The slope-intercept form of a line's equation is widely used and highly intuitive. It's written as \(y = mx + b\), where:

• \(m\) represents the slope
• \(b\) represents the y-intercept, the point where the line crosses the y-axis

Starting from the point-slope form, we simplify:

• \( y + 4 = -\frac{1}{2}x - 1 \)
• \( y = -\frac{1}{2}x - 5 \)

Thus, in slope-intercept form, \(-\frac{1}{2}\) is the slope and \(-5\) is the y-intercept.
This form makes it easy to graph the line as it vividly shows the slope and y-intercept.

Coordinate Geometry

Coordinate geometry combines algebra and geometry to study geometric figures using the coordinate plane. It allows representing geometric shapes algebraically and helps explore properties such as distance, midpoints, and slopes.
In our exercise, we use coordinate geometry principles to formulate equations of a straight line based on given points:

• The slope between points offers a clue about the line's direction.
• Various line forms, such as point-slope and slope-intercept, emerge from this understanding.
• Finally, we convert equations into relevant forms to meet specific needs, like scaling or simplifying.

Coordinate geometry provides the tools needed to go from the visual to the analytical, improving comprehension of geometric relationships.