Problem 53 - Algebra and Trigonometry

Question

Write the linear system whose solution set is {(6, 2)}. Express each equation in the system in slope-intercept form.

Step-by-Step Solution

Answer

The linear system that has a solution at (6,2) is given by the equations y=2, x=6.

Step 1: Select First Line

Choose a line that passes through the point (6,2). For the first line, we can choose a slope of 0 meaning this is a horizontal line and it will pass through any point whose y-coordinate is 2. This gives the equation y=2.

Step 2: Select Second Line

Choose a different line that also passes through the point (6,2). For simplicity, choosing a vertical line which passes through the points whose x-coordinate is 6. This line can be expressed in the form x=6.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most recognizable and useful representations in algebra. It is given by the formula \( y = mx + b \). In this formula:

• \( y \) represents the dependent variable, often corresponding to the y-coordinate on a graph.
• \( m \) is the slope of the line, which indicates how steep the line is.
• \( x \) is the independent variable, often corresponding to the x-coordinate on a graph.
• \( b \) is the y-intercept, or the point where the line crosses the y-axis.

Understanding this form means you can easily determine both the slope and the y-intercept of any line. For instance, if you had the equation \( y = 3x + 1 \), the slope \( m \) would be 3, and the y-intercept \( b \) would be 1.

Solution Set

A solution set in mathematics refers to all possible solutions for an equation or system of equations. When you solve a linear system, like in the original exercise where the solution set is \((6, 2)\), it means that both equations intersect exactly at that point.

This means the point satisfies all the equations simultaneously. Understanding the solution set is crucial when dealing with systems of equations as it provides the exact values of \( x \) and \( y \) that solve the system. Often, these solutions can be visualized on a graph as the point of intersection of lines.

Horizontal Line

A horizontal line is one that runs left to right across your graph and remains unchanged in value, regardless of any x-coordinate. It is expressed in the slope-intercept form as \( y = b \). This type of line has a slope \( m \) equal to 0, meaning it does not rise or fall as x changes.

For example, in our original exercise, the horizontal line is \( y = 2 \), implying all points on this line have a y-coordinate of 2. This means that any variation in x does not affect y, resulting in a flat, straight line on a graph.

Vertical Line

Vertical lines are those that go up and down on a graph and remain unchanged in value, regardless of any y-coordinate. Unlike horizontal lines, a vertical line does not fit into the slope-intercept form because its slope is undefined. Instead, a vertical line is represented by \( x = a \), where \( a \) is the constant x-value for that line.

In the given exercise, the vertical line is \( x = 6 \), indicating all points on this line share an x-coordinate of 6. While the y-coordinates can vary, the x-value remains constant, creating a line that shoots straight up and down on the graph.