Problem 46 - Elementary Algebra

Question

For Problems \(45-60\), write the equation of the line that satisfies the given conditions. Express final equations in standard form. \(x\) intercept of \(-1\) and \(y\) intercept of \(-3\)

Step-by-Step Solution

Answer

The equation in standard form is \(3x + y = -3\).

Step 1: Understand Intercepts

The x-intercept of -1 means the line crosses the x-axis at the point (-1,0), and the y-intercept of -3 means it crosses the y-axis at the point (0,-3).

Step 2: Use Two-Point Formula

With points (-1, 0) and (0, -3), calculate the slope (m) using the formula: \( m = \frac{y_2-y_1}{x_2-x_1} \). This gives \( m = \frac{-3-0}{0-(-1)} = -3 \).

Step 3: Write Slope-Intercept Form

Using the slope -3 and y-intercept (0, -3), the slope-intercept form is \( y = -3x - 3 \).

Step 4: Convert to Standard Form

Rearrange the equation \( y = -3x - 3 \) to standard form, \( Ax + By = C \) by moving all terms to one side: \(3x + y = -3\).

Key Concepts

x-intercept

The x-intercept of a line is the point where the line crosses the x-axis. This means that at the x-intercept, the y-coordinate is always 0. In our example, the x-intercept is given as (-1, 0). This point indicates that when you move along the line, it will touch the x-axis at x = -1.
Understanding the x-intercept is crucial because it helps describe the position of the line on a graph. Moreover:

• The x-intercept can be found by setting y to 0 in the line equation and solving for x.
• It provides useful information for sketching the line quickly.

Knowing the x-intercept, along with the slope, allows for writing the line's equation, which can then be transformed into any desired form, such as the slope-intercept form or standard form.

y-intercept

The y-intercept is the point where the line crosses the y-axis, which means at this point, the x-coordinate is always zero. For our problem, the y-intercept is at (0, -3), indicating that the line touches the y-axis at y = -3.
This concept is essential because it gives us an anchor point on the graph where the line will pass through. Some important aspects include:

• At the y-intercept, you set x to 0 in the equation and solve for y.
• This helps in writing the slope-intercept equation directly as it's the constant term.

With the y-intercept known, along with another point or the slope, the equation of the line can be developed and manipulated to any desired form.

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 non-negative. It provides a structured way to present line equations and is particularly useful in specific calculations involving solving systems of equations or identifying parallel or perpendicular lines.
To convert a line equation into standard form, like in our example, follow these steps:

• Ensure that all terms involving x and y are on the same side of the equation.
• Rearrange the equation by moving the x-term to the left and adjusting other terms accordingly, ensuring integer coefficients.
• In our case, \(y = -3x - 3\) becomes \(3x + y = -3\).

Converting to standard form allows easier interpretations and solutions that involve multiple equations.

slope-intercept form

The slope-intercept form of a linear equation is expressed as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) represents the y-intercept. This form is very intuitive and easy to use for graphing, as it directly shows the line's tilt (slope) and where it crosses the y-axis (y-intercept).
For our exercise, the slope was determined as \(m = -3\) from the changes in y and x, and with the y-intercept being \(-3\), this directly gives us the equation \(y = -3x - 3\). Here are some pertinent features:

• The "m" value shows how steep the line is and its direction.
• The "b" value provides the exact point where the line crosses the y-axis.

Using the slope-intercept form simplifies the process of sketching and understanding the behavior and trajectory of the line immediately from its equation.