Problem 48 - Algebra 1

Question

Write an equation of the line that passes through the points. (-2,-3),(0,4)

Step-by-Step Solution

Answer

The equation of the line that passes through the points (-2,-3) and (0,4) is \(y = 3.5x + 4\).

Step 1: Calculate the slope

The slope of the line, \(m\), can be determined using the two given points and the following formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points (-2,-3) and (0,4) into the formula gives \(m = \frac{4 - (-3)}{0 - (-2)} = \frac{7}{2} = 3.5\). Therefore, the slope of the line is 3.5.

Step 2: Find the y-intercept

Substitute one of the given points and the slope into the equation \(y = mx + c\) to solve for \(c\), the y-intercept. Using the point (0,4), the equation becomes 4 = 3.5*0 + c, which simplifies to \(c = 4\). Therefore, the y-intercept is 4.

Step 3: Write the equation of the line

The equation of the line in slope-intercept form is \(y = mx + c\). Substituting the values determined in earlier steps, the equation therefore is \(y = 3.5x + 4\).

Key Concepts

Slope of a Line

In understanding linear equations, a crucial concept is the "slope of a line." The slope denotes how steep a line is and the direction in which it slopes. It is represented by the letter \( m \), and its value can be calculated by comparing the rise (change in \( y \)-values) to the run (change in \( x \)-values) between two points on the line.

When you have two points \((-2, -3)\) and \((0, 4)\), you can calculate the slope using the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1} \]Where \( (x_1, y_1) = (-2, -3) \) and \( (x_2, y_2) = (0, 4) \). This formula will give you:
\[m = \frac{4 - (-3)}{0 - (-2)} = \frac{7}{2} = 3.5 \]

• The numerator (4 - (-3)) calculates the vertical difference or rise, which is \( 7 \).


• The denominator (0 - (-2)) calculates the horizontal difference or run, which is \( 2 \).

From this, we see the slope of the line is \( 3.5 \), indicating a positive slope that rises steeply.

Equation of a Line

Once the slope of a line is known, the next step is forming the "equation of a line." An equation tells us all the possible points lying on a line. Generally, a line in the Cartesian coordinate system is described by the format \( y = mx + c \), where:

• \( m \) is the slope.


• \( c \) is the y-intercept.

To find the value of \( c \), substitute a point that lies on the line along with the known slope into the equation.
Using the point \((0, 4)\) and the slope \( m = 3.5 \), the equation becomes:
\[y = 3.5 \times 0 + c = 4 \]This step helps determine that \( c = 4 \), so the line crosses the \( y \)-axis at \( (0, 4) \).

Knowing both \( m \) and \( c \) allows us to configure the equation that describes every point on the line effectively.

Slope-Intercept Form

One of the most common forms used to write the equation of a line is the "slope-intercept form." This form is expressed as: \[y = mx + c \]Here, \( m \) represents the slope, while \( c \) is the y-intercept.

The slope-intercept form is particularly useful for quickly illustrating the properties of a line, such as how it travels across the graph and where it intersects the y-axis. Consider the scenario where the slope \( m \) is calculated as \( 3.5 \) and the y-intercept \( c \) is determined as \( 4 \). Filling these values in gives us the equation:
\[y = 3.5x + 4 \]

• This tells us that for every unit increase in \( x \), \( y \) increases by \( 3.5 \).


• The equation confirms that the line meets the y-axis right at \( 4 \).

Utilizing the slope-intercept form simplifies understanding and drawing the line on a graph. It provides clear insight intokey characteristics such as slope and intercept, making it a vital toolin linear algebra.