Problem 16 - Algebra and Trigonometry

Question

\(15-20\) m Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$ \left\\{\begin{array}{l}{2 x-y=4} \\ {3 x+y=6}\end{array}\right. $$

Step-by-Step Solution

Answer

The system has one solution at the point \((2, 0)\).

Step 1: Convert each equation to slope-intercept form

For the first equation, \(2x - y = 4\), solve for \(y\): \(y = 2x - 4\). For the second equation, \(3x + y = 6\), solve for \(y\): \(y = -3x + 6\).

Step 2: Identify slopes and y-intercepts

The first equation \(y = 2x - 4\) has a slope \(m = 2\) and a y-intercept \(b = -4\).The second equation \(y = -3x + 6\) has a slope \(m = -3\) and a y-intercept \(b = 6\).

Step 3: Graph both lines on the coordinate plane

Plot the y-intercept of the first line at \((0, -4)\) and use the slope \(2/1\) to find another point by moving up 2 and right 1.Plot the y-intercept of the second line at \((0, 6)\) and use the slope \(-3/1\) to find another point by moving down 3 and right 1. Draw both lines.

Step 4: Determine the point of intersection

Observe where the two lines cross on the graph. The intersection occurs at the point \((2, 0)\).

Step 5: Confirm the solution by substitution

Substitute \(x = 2\) and \(y = 0\) into both original equations to verify:1) \(2(2) - 0 = 4\) is true.2) \(3(2) + 0 = 6\) is true.The point \((2, 0)\) satisfies both equations.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is given as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) denotes the y-intercept, the point where the line crosses the y-axis.

This form is particularly user-friendly for graphing because it directly provides key information about the line:

• The slope \(m\) indicates the steepness and direction of the line. If \(m\) is positive, the line rises as it moves from left to right; if negative, the line falls.
• The y-intercept \(b\) shows where the line will touch the y-axis, making it an ideal starting point for plotting the line on a graph.

By converting a linear equation into this form, you ensure an easy pathway for visualizing the line and understanding its behavior. With simple algebraic manipulation, you can transform standard equations, like \(2x - y = 4\) and \(3x + y = 6\), into the slope-intercept form to facilitate graphing and analysis. This method not only makes the graphing process straightforward but also offers a clearer insight into how changes in equations affect their corresponding lines.

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane and drawing a straight line through them. The process begins by identifying the slope and y-intercept from the slope-intercept form, \(y = mx + b\).

To graph an equation:

• Start by plotting the y-intercept on the y-axis. For example, in \(y = 2x - 4\), place a point at \(0, -4\).
• Next, use the slope to find additional points. A slope of \(2\) means moving up 2 units and right 1 unit from the y-intercept. Repeat this step to trace the path of the line.
• Draw a straight line through the points to complete the graph.

Following this method, you can graph the second equation, \(y = -3x + 6\), similarly, beginning at the y-intercept \(0, 6\) and using the slope \(-3\) to move down 3 units and right 1 unit.

Graphing helps visualize solutions to systems of equations and provides a tangible way to understand the relationships between equations.

Point of Intersection

In a system of linear equations, the point of intersection is where the graphs of the equations meet. This point represents the solution to the system and indicates values of \(x\) and \(y\) that satisfy both equations simultaneously.

To find the point of intersection:

• Graph each equation on the same coordinate plane as detailed in previous sections.
• Identify where the lines cross.

In this example, the lines cross at \(2, 0\), meaning \(x = 2\) and \(y = 0\) is the solution to the system.

After locating this intersection, it's essential to verify by substituting these values into both original equations: For \(2x - y = 4\): \(2(2) - 0 = 4\) holds true, and for \(3x + y = 6\): \(3(2) + 0 = 6\) also holds true. Both checks confirm \(2, 0\) is indeed the correct solution. Understanding the point of intersection enhances your comprehension of how linear equations relate and their solutions intersect within a graph.