Problem 53 - Precalculus with Limits

Question

In Exercises 51-64, find the slope-intercept form of the equation of the line that passes through the given point and has the indicated slope \(m\). Sketch the line. \((-3, 6)\), \(m = -2\)

Step-by-Step Solution

Answer

The slope-intercept form of the equation of the line that passes through the point (-3,6) and has the slope -2 is \(y = -2x\). The line descends at a slope of -2 and crosses through the origin (0,0).

Step 1: Insert the given values into the slope-intercept formula

Substitute the \(x\), \(y\), and \(m\) values, \((-3,6)\) and \(-2\), into the formula \(y = mx + c\).\nHence, \[6 = -2*(-3) + c\]

Step 2: Solve for the y-intercept \(c\)

Solve the equation \[6 = -2*(-3) + c\] for \(c\).\nAfter calculation, we get \(c = 0\).

Step 3: Write the final equation in slope-intercept form

Substitute \(m = -2\) and \(c = 0\) into the equation \(y = mx + c\).\nThe final slope-intercept form of the equation is \(y = -2x\)

Step 4: Sketch the line

Sketch the line based on the resulting equation from the previous step.\nThis line has a slope of \(-2\), which means for every increment of \(1\) on the \(x\)-axis, the line descends by \(2\) on the \(y\)-axis.\nAlso, the line crosses through the origin (0,0), because the y-intercept \(c = 0\).

Key Concepts

Linear Equations

A linear equation is one of the fundamental concepts in algebra and represents a straight line in a two-dimensional space. It typically takes on the form of \( y = mx + c \) where \( m \) stands for the slope of the line and \( c \) represents the y-intercept, which is the point where the line crosses the y-axis. The slope tells us how steep the line is, and the y-intercept gives us its exact position on the graph.

Understanding linear equations is crucial because they're used to model real-world situations where there is a constant rate of change. For example, if you're calculating the cost of apples where each apple costs $2, the total cost would be a linear equation relating the number of apples bought and the total price you pay.

Graphing Lines

Graphing a line on the coordinate plane is a visual way to understand the relationship between two variables. When given a linear equation, like in our example \( y = -2x \) from the slope-intercept form, you start by plotting the y-intercept. In this case, the y-intercept is at the origin (0,0) since \( c = 0 \).

After plotting the y-intercept, use the slope, which is \( -2 \) in our example, to determine the direction of the line. The slope tells us to move 2 units down for every one unit we move to the right, because the slope is negative. By continuing this pattern from the y-intercept, we can draw the line that represents the equation.

An important tip for students is to remember that two points are enough to draw a straight line. However, plotting a third point as a confirmation can help ensure accuracy in your graph.

Slope and Y-Intercept

The slope and y-intercept are key components of the slope-intercept form of a linear equation, which enables you to understand and graph the line quickly. The slope, represented by \( m \) in the equation \( y = mx + c \) is a measure of the line's steepness and direction. A positive slope goes upwards from left to right, whereas a negative slope, like \( -2 \) in our example, goes downwards.

The y-intercept, represented by \( c \) in the equation, is the coordinate where the line crosses the y-axis. This is the starting point when drawing a line on a graph. When \( c \) is zero, the line passes through the origin.

If you're struggling to grasp these concepts, try using a table of values. This method involves choosing a few values for \( x \), plugging them into the equation to find the corresponding \( y \) values, and plotting the resulting points on the graph to create the line. This can often make the process of graphing more tangible and easier to understand.