Problem 45 - Elementary Algebra

Question

Write the equation of the line that satisfies the given conditions. Express final equations in standard form. \(x\) intercept of 2 and \(y\) intercept of \(-4\)

Step-by-Step Solution

Answer

The equation is \(2x - y = 4\) in standard form.

Step 1: Understanding the Intercepts

To write the equation of a line, we need to understand the information given. The problem provides an x-intercept of 2 and a y-intercept of -4. This means the line passes through the points (2, 0) and (0, -4).

Step 2: Finding the Slope

Using the points (2, 0) and (0, -4), the slope can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute \( (x_1, y_1) = (2, 0) \) and \( (x_2, y_2) = (0, -4) \), the equation becomes:\[ m = \frac{-4 - 0}{0 - 2} = \frac{-4}{-2} = 2 \]

Step 3: Formulating the Equation in Slope-Intercept Form

With the slope \( m = 2 \) and using the y-intercept \( b = -4 \), the slope-intercept form \( y = mx + b \) becomes:\[ y = 2x - 4 \]

Step 4: Rewriting in Standard Form

The standard form of a linear equation is \( Ax + By = C \). To convert \( y = 2x - 4 \) to standard form, rearrange the equation:Subtract \( 2x \) from both sides:\[ -2x + y = -4 \]Multiplying by -1 to keep coefficients positive gives:\[ 2x - y = 4 \]

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most commonly used forms because it directly tells us the slope and y-intercept of the line. It's written as \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept—the value of \( y \) when \( x \) is 0.

This form is especially useful when you need to quickly graph a line or understand its behavior. For example, a line with a slope of 2 and a y-intercept of -4 can be expressed as \( y = 2x - 4 \). This tells us:

• The line rises by 2 units vertically for every 1 unit it moves horizontally.
• It crosses the y-axis at -4.

Graphically, starting at the point (0, -4), you would rise 2 units up and move 1 unit to the right to find another point on the line. Repeating this, you can draw the line accurately. Understanding the slope-intercept form is crucial for various applications in mathematics and real-world problem solving.

Standard Form

The standard form of a linear equation is expressed as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and typically, \( A \) should be a positive integer. This form is particularly useful when solving systems of equations or when you need the equation to be neat and integer-based.

To convert from slope-intercept form to standard form, follow these steps:

• Move the \( x \)-term to the left side of the equation.
• Adjust the equation so that \( A \) is positive.
• Make sure all coefficients are integers.

For instance, converting \( y = 2x - 4 \) would involve subtracting \( 2x \) from both sides to get \( -2x + y = -4 \). Multiplying everything by -1 results in \( 2x - y = 4 \), which is the standard form. This format streamlines solving equations, particularly when dealing with more complex problems or involving multiple equations at once.

Coordinate Geometry

Coordinate Geometry, also known as Analytic Geometry, allows us to use algebra to solve geometric problems. This branch of geometry is used extensively to determine the position of points, lines, and shapes on the coordinate plane.

By using coordinate geometry, you can find lengths, angles, and various other properties of geometric figures without requiring traditional, sometimes complex, geometrical methods.

For the given problem, the x-intercept of 2 and y-intercept of -4 directly translate to the points (2, 0) and (0, -4) on a 2D plane.

This understanding of points, lines, and intercepts forms the basis of many real-world calculations, such as finding the shortest distance between points or determining intersection points between lines and curves. Coordinate geometry not only simplifies mathematical visualization but also amplifies its computational power, making it an essential tool in various scientific fields.