Problem 74 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line that passes through the given points. $$ (-4,-5) \text { and }(0,3) $$

Step-by-Step Solution

Answer

The equation of the line passing through the points (-4,-5) and (0,3) is \(y=2x+3\) .

Step 1: Compute the Slope

Determine the slope \(m\) of the line using the given points (-4,-5) and (0,3). The formula to calculate slope is given by \[ m = (y_2 - y_1) / (x_2 - x_1) \], where \( (x_1, y_1) = (-4, -5) \) and \( (x_2, y_2) = (0, 3) \). Therefore, \[ m = (3 - (-5)) / (0 - (-4)) = 8 / 4 = 2 \].

Step 2: Calculate y-Intercept

Once the slope is determined, find y-intercept \(c\). By substituting one of the points (use (0,3) here because \(c\) is the value where x=0) and the slope value \(m\) into the equation \(y = mx + c\), solve for \(c\). By substituting the value of slope \(m=2\) and the point (0,3), you get \(3 = (2 * 0) + c\), thus \(c = 3\) .

Step 3: Write Equation in Slope-Intercept Form

Now, write out the equation of the line in slope-intercept form by substituting values of \(m\) and \(c\) into the equation \(y=mx+c\). Thus, your equation is \(y=2x+3\) . This is the equation of the line that passes through the points (-4,-5) and (0,3)

Key Concepts

Linear Equations

Linear equations are an essential part of algebra and are frequently encountered in mathematics. They represent straight lines when graphed on a coordinate plane. A linear equation’s standard form is \( Ax + By = C \). However, a more convenient form for students and applications is the slope-intercept form \( y = mx + b \), where \( y \) is the dependent variable and \( x \) is the independent variable. The variables \( m \) and \( b \) are constants representing the slope and y-intercept of the line, respectively.
The slope-intercept form is useful because it directly gives you useful information about the line. The equation becomes straightforward to graph since you can easily identify where the line crosses the y-axis and how steep the line is. This form is especially handy for quickly identifying lines’ characteristics without needing to rearrange or manipulate the equation.
When dealing with problems related to this form, always remember:

• Identify the slope \(m\).
• Determine the y-intercept \(b\).
• Write the equation in the form \(y = mx + b\).

Slope Calculation

The slope of a line is a measure of how steep the line is. In simple terms, it tells us how much \( y \) changes for a change in \( x \). Mathematically, the slope \( m \) is calculated using the formula:

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

Here,

• \((x_1, y_1)\) and \((x_2, y_2)\) are any two distinct points on the line.

In our problem, the points were \((-4, -5)\) and \((0, 3)\). By substituting these into the formula, we found that the slope \( m = 2 \). This means that for every unit increase in \( x \), \( y \) increases by 2 units.
Remember:

• If \( m > 0 \), the line is inclined upwards (positive slope).
• If \( m < 0 \), the line declines downwards (negative slope).
• If \( m = 0 \), the line is horizontal (zero slope).

Y-Intercept

The y-intercept is another key component of understanding linear equations. It represents the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the value \( b \) is the y-intercept. It tells us the value of \( y \) when \( x \) is zero.
Finding the y-intercept is a straightforward process in many exercises:

• Set \( x = 0 \) in the equation.
• Solve for \( y \) which will give the y-intercept \( b \).

In the step-by-step solution we have, you see this process: by substituting \((0, 3)\) into the equation along with the slope, it was confirmed that the y-intercept \( c = 3 \).
It’s important to remember that:

• The y-intercept allows for quick graph plotting.
• It provides a starting point on the graph where the value of \( x \) is zero.

Understanding the y-intercept helps in visualizing the full behavior of linear relationships and serves as a critical foundation in solving algebraic problems involving lines.