Problem 58 - Algebra 2

Question

Write an equation in slope-intercept form that satisfies each set of conditions. passes through \((0,-2)\) and \((4,2)\)

Step-by-Step Solution

Answer

The equation is \( y = x - 2 \).

Step 1: Identify Slope Formula

To write an equation in slope-intercept form, which is \( y = mx + b \), we first need to find the slope \( m \). The slope (abla) between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step 2: Plug in Given Points

Substitute the given points \((0,-2)\) and \((4,2)\) into the slope formula: \[ m = \frac{2 - (-2)}{4 - 0} = \frac{2 + 2}{4} = \frac{4}{4} = 1 \]. Thus, the slope \( m \) is 1.

Step 3: Setup Equation in Slope-Intercept Form

Now that we know \( m = 1 \), we can use the point-slope form, \( y - y_1 = m(x - x_1) \), with \((0, -2)\) to find \( b \). However, since \((0, -2)\) already provides \( b \) directly as -2, we note \( b = -2 \) and move on to write the final equation.

Step 4: Write the Final Equation

The slope-intercept form of the line is \( y = mx + b \). With \( m = 1 \) and \( b = -2 \), the equation becomes \( y = 1x - 2 \) or simply \( y = x - 2 \).

Key Concepts

Calculating Slope

To understand the equation of a line in the slope-intercept form, it's crucial to start with calculating the slope. The slope of a line is essentially a measure of its steepness. It tells us how much the line rises or falls as we move along the x-axis.

To calculate the slope, represented as "m" in the equation, we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula uses two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line. The difference between the \( y \) values (rise) divided by the difference between the \( x \) values (run) gives the slope. For example, if we take points \((0,-2)\) and \((4,2)\), applying the formula:

• Calculate the rise: \( 2 - (-2) = 4 \)
• Calculate the run: \( 4 - 0 = 4 \)
• Slope \( m = \frac{4}{4} = 1 \)

A slope of 1 tells us that for every unit we move to the right, the line rises by 1 unit.

Linear Equations

Linear equations are equations of the first degree, meaning that the variable(s) are raised only to the power of 1. In the context of two-dimensional graphing, a linear equation represents a straight line. One common form to represent these equations is the slope-intercept form: \( y = mx + b \) Here, "m" represents the slope, and "b" is the y-intercept—the point where the line crosses the y-axis.

To formulate a linear equation from points, we use the slope and one of the points to determine the y-intercept. In our exercise, we found the slope to be 1, and since we also know a point on the line \((0, -2)\), we can directly see the value of "b" as \(-2\). Thus, the equation is: \( y = 1x - 2 \) This can be simplified to \( y = x - 2 \). This shows that the line crosses the y-axis at \((0, -2)\).

Graphing Lines

Graphing a line based on its equation in slope-intercept form, \( y = mx + b \), is straightforward. Start by identifying the y-intercept "b". This is where the line will cross the y-axis.

For the equation \( y = x - 2 \), the y-intercept is \(-2\). Plot this point at \( (0, -2) \) on the coordinate graph. With the slope \( m = 1 \), you know that for each unit you move right, you move up one unit.

• Start at the y-intercept \((0, -2)\).
• From there, move right 1 unit and up 1 unit to plot another point \((1, -1)\).
• Continue this pattern or, equivalently, move left 1 unit and down 1 unit to extend the line if needed.

By connecting the points, you form a complete line that represents the equation \( y = x - 2 \). Any point on this line satisfies the linear equation because it has the same slope and intercept, confirming the correctness of your graph.