Problem 41 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. Through (10,7) and (7,10)

Step-by-Step Solution

Answer

The equation of the line is \(y = -x + 17\).

Step 1: Understand the Problem

We need to find the equation of a line that passes through the points \((10,7)\) and \((7,10)\). We will express this equation in the slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Find the Slope

To find the slope \(m\) of the line, use the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substituting the coordinates of the points: \[m = \frac{10 - 7}{7 - 10} = \frac{3}{-3} = -1\]So, the slope \(m\) of the line is \(-1\).

Step 3: Use the Point-Slope Form

Now that we have the slope, use the point-slope form of a line equation: \[y - y_1 = m(x - x_1)\]Choose a point, say \((10,7)\). Substituting \(m = -1\), \(x_1 = 10\), and \(y_1 = 7\) into the equation: \[y - 7 = -1(x - 10)\]

Step 4: Simplify to Slope-Intercept Form

Distribute \(-1\) on the right-hand side:\[y - 7 = -x + 10\]Add 7 to both sides to solve for \(y\):\[y = -x + 10 + 7\]\[y = -x + 17\]Thus, the equation of the line in slope-intercept form is \(y = -x + 17\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a very user-friendly way to write the equation of a line. This format is expressed as \(y = mx + b\). Here, \(m\) represents the slope of the line, which tells you how steep the line is, and \(b\) is the y-intercept, where the line crosses the y-axis.

• Slope - it indicates the direction and steepness of the line. A positive slope means the line is rising, while a negative slope means the line is falling.
• Y-intercept - this is the point on the graph where the line crosses the y-axis. It's the value of \(y\) when \(x\) is zero.

The convenience of slope-intercept form is that it allows you to easily visualize and graph the line just by knowing two things: its slope and where it meets the y-axis. To convert a line equation into slope-intercept form, you merely need to solve for \(y\) in terms of \(x\), which is very handy when graphing lines.

Point-Slope Form

The point-slope form is another approach to writing the equation of a line. This form is very helpful when you know a point on the line and the slope. The equation is given by:\[ y - y_1 = m(x - x_1) \]Where \((x_1, y_1)\) are the coordinates of a given point on the line, and \(m\) is the slope.

• Use this form when you have one point and the slope. This makes it ideal for situations when you're given a specific point through which the line passes and the line's slope.

To convert it into slope-intercept form, you just need to solve the equation for \(y\). This involves distributing and then isolating \(y\). Point-slope form is particularly valuable because it can quickly lead you to the slope-intercept form, especially with its handy substitution method.

Finding Slope

Finding the slope of a line is an essential skill in algebra, particularly when dealing with linear equations. The slope tells you how the line changes, or "moves," as you move along from point to point. You can determine the slope using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Via this formula, you're measuring the change in \(y\) (the vertical change) over the change in \(x\) (the horizontal change) between two points. Let's break this down:

• Choose any two points on the line, labeled \(x_1, y_1\) and \(x_2, y_2\).
• Subtract the y-values of these points to find the change in the vertical direction.
• Subtract the x-values of these points to find the change in the horizontal direction.
• Divide the change in \(y\) by the change in \(x\).

A very consistent slope means your line will have constant steepness. Remember, a positive slope slopes upwards from left to right, and a negative slope slopes downwards. A zero slope is flat, depicting a horizontal line, while an undefined slope reflects a vertical line. It's a powerful tool to describe how lines move through coordinate planes.