Problem 33 - Algebra A Combined Function

Question

Find an equation of the line passing through each pair of points. Write the equation in the form $A x+B y=C. $$ (-1,3) \text { and }(-2,-5) $$

Step-by-Step Solution

Answer

The equation of the line is \(8x - y = -11\).

Step 1: Calculate the Slope

To find the equation of a line, we first need to calculate the slope (), which is the rise over the run between the two points. Use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1) = (-1, 3)\) and \((x_2, y_2) = (-2, -5)\). Substitute the values into the formula to get \(m = \frac{-5 - 3}{-2 - (-1)} = \frac{-8}{-1} = 8\).

Step 2: Use Point-Slope Form

With the slope \(m = 8\) and a point \((-1, 3)\), use the point-slope form of the line equation: \(y - y_1 = m(x - x_1)\). Substitute \(m = 8\), \(x_1 = -1\), and \(y_1 = 3\) into the formula: \(y - 3 = 8(x + 1)\).

Step 3: Simplify the Equation

Expand the equation \(y - 3 = 8(x + 1)\) to get \(y - 3 = 8x + 8\). To simplify, add 3 to both sides: \(y = 8x + 11\).

Step 4: Convert to Standard Form

To write the equation \(y = 8x + 11\) in standard form \(Ax + By = C\), rearrange it as \(-8x + y = 11\), which becomes our final equation: \(8x - y = -11\). Multiply through by -1 to avoid a negative \(A\) and get \(-8x + y = 11\).

Step 5: Finalize the Standard Form

Rearrange one more time if needed to make sure coefficients are integers: \(8x - y = -11\).

Key Concepts

Slope Calculation

Understanding how to calculate the slope of a line is a fundamental concept in algebra. The slope describes how steep a line is. It is the ratio of the rise (the change in the y-values) to the run (the change in the x-values) between two points.To find the slope, use the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Where:

• \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line.
• The difference in the y-values \((y_2 - y_1)\) is known as the rise.
• The difference in the x-values \((x_2 - x_1)\) is referred to as the run.

For the points \((-1, 3)\) and \((-2, -5)\), the slope is calculated as:\[ m = \frac{-5 - 3}{-2 - (-1)} = \frac{-8}{-1} = 8 \]This indicates that for each unit increase in x, the y-value increases by 8.

Point-Slope Form

After determining the slope, the next step is to use the point-slope form to express the line's equation. This form is especially useful because it incorporates both a point on the line and the slope.The point-slope form equation is:\[ y - y_1 = m(x - x_1) \]Where:

• \(m\) is the slope of the line.
• \((x_1, y_1)\) is a given point on the line.

For our line with a slope \(m = 8\) and passing through the point \((-1, 3)\), the equation becomes:\[ y - 3 = 8(x + 1) \]The point-slope form is often an intermediate step before converting the equation to other, more familiar forms, such as the slope-intercept form.

Standard Form of a Line

The standard form of a linear equation is one of the three primary forms you might use when expressing equations of lines. It has the general structure:\[ Ax + By = C \]Where:

• \(A, B,\) and \(C\) are integers.
• \(A\) should be a positive integer.

To convert the equation from the point-slope form, simplify and rearrange the equation slightly. Starting with:\[ y - 3 = 8(x + 1) \]This can be expanded to:\[ y - 3 = 8x + 8 \]Add 3 to both sides to simplify:\[ y = 8x + 11 \]Rearranging it to the standard form gives:\[ -8x + y = 11 \]We usually prefer \(A\) to be positive, so multiply the whole equation by -1:\[ 8x - y = -11 \]

Linear Equations

Linear equations represent a straight line when graphed on a coordinate plane. They have various forms, including slope-intercept, point-slope, and standard forms. Each form has its particular use based on the information given or needed.Important aspects of linear equations include:

• The slope, which identifies the steepness of the line. A positive slope means the line rises as it moves to the right, while a negative slope means it falls.
• Intercepts where the line crosses the axes. The y-intercept is the point where the line crosses the y-axis, found in equations in the slope-intercept form as "\(b\)" in \(y = mx + b\).
• The ability to express linear relationships in situations like Constant Speed, Budgeting, and other real-life applications.

Grasping how to move between different forms of linear equations and understanding what each component represents opens a wide array of problem-solving strategies in algebra.