Problem 33 - Algebra 1

Question

Write an equation in standard form of the line that passes through the given point and has the given slope. $$(-8,3), m=2$$

Step-by-Step Solution

Answer

Thus, the standard equation of a line that passes through the point (-8,3) and has a slope of 2 is \(-2x + y = 19\).

Step 1: Construct the Equation in the Slope-Intercept Form

In the slope-intercept form (y = mx + b), the given slope (m) is 2 and the coordinate points (-8,3) can be substituted for x and y separately. Therefore, the equation would look like this: 3 = 2*(-8) + b After simplifying, b is calculated as b = 3 - 2*(-8) = 3 + 16 = 19.

Step 2: Complete the Equation in the Slope-Intercept Form

With the slope (m) and y-intercept (b) established, the equation in slope-intercept form can be completed as:y = 2x + 19

Step 3: Convert to Standard Form

After that, y = 2x + 19 has to be changed to the standard form (Ax + By = C). When shifting the terms around, it is critical to make sure that the coefficient in front of x is a positive integer number:-2x + y = 19

Key Concepts

Standard Form

The standard form of a linear equation is expressed as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) is a non-negative integer. This form is particularly useful because it clearly displays the relationship between the \(x\) and \(y\) coordinates.

To convert an equation into this form, you may need to rearrange terms or adjust coefficients so that \(A\) is positive. In the context of our original exercise, the equation received in slope-intercept form was \(y = 2x + 19\). To transform this into standard form:

• Subtract \(2x\) from both sides to align terms involving \(x\) and \(y\): \(-2x + y = 19\).
• To ensure \(A\) is positive, multiply the entire equation by -1, resulting in the standard form: \(2x - y = -19\).

This concise form is optimal for quickly comparing lines and determining their intercepts.

Slope-Intercept Form

The slope-intercept form, \(y = mx + b\), is highly popular for expressing linear equations. Here, \(m\) symbolizes the slope of the line, while \(b\) represents the y-intercept, the point where the line crosses the y-axis. Using slope-intercept form can make it easier to graph a line or understand its behavior visually.

In solving the original exercise, the slope-intercept form was applied initially to shape the equation. Given the slope \(m = 2\) and a point on the line, \((-8, 3)\), the form allows us to substitute \(x = -8\) and \(y = 3\):

• Start with the equation: \(y = mx + b\).
• Substitute the given values: \(3 = 2 \times (-8) + b\).
• Simplify to find the intercept \(b\): \(b = 3 + 16 = 19\).

Ultimately, the slope-intercept form of the equation becomes \(y = 2x + 19\), illustrating both the slope and the intercept.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics provides a powerful connection between algebra and geometry through the use of coordinates and equations to represent geometric figures.

In coordinate geometry, points are expressed using coordinates \((x, y)\). These coordinates allow us to understand and describe the position of the points on a plane. Lines can be represented as equations in different forms, including the standard and slope-intercept forms.

In the context of our problem, the point \((-8, 3)\) indicates a specific location on the Cartesian plane. When combined with the slope \(m = 2\), it enables the formulation of a specific line.

• You can visualize this line passing through the given point with the specified slope.
• Analyzing the equation graphically allows us to understand the geometric relationship and transformations on the plane fully.

Thus, coordinate geometry binds algebraic equations and geometric images, enhancing our ability to solve and interpret mathematical problems in a visual space.