Problem 38 - Algebra and Trigonometry

Question

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. \(\left\\{\begin{array}{l}{y=-2 x+12} \\ {y=x+3}\end{array}\right.\)

Step-by-Step Solution

Answer

The lines intersect at (3, 6), which is the solution.

Step 1: Understand the System of Equations

We have a system of linear equations given by: 1. \( y = -2x + 12 \)2. \( y = x + 3 \)The objective is to find the solution of this system using the graphical method, which is the intersection point of these two lines.

Step 2: Graph the First Equation

Rewrite the first equation \( y = -2x + 12 \) in slope-intercept form. The slope is \(-2\) and the y-intercept is \(12\). Plot the y-intercept on the graph, then use the slope to plot another point (move down 2 units and right 1 unit). Draw the line through these points.

Step 3: Graph the Second Equation

The second equation \( y = x + 3 \) is already in slope-intercept form, with slope \(1\) and y-intercept \(3\). Start by plotting the y-intercept. Then, from that point, move up 1 unit and right 1 unit to plot another point. Draw the line through these points to extend it.

Step 4: Find the Intersection Point

Look at the graph and identify the point where the two lines intersect. The intersection point gives us the solution to the system of equations, where both equations are satisfied simultaneously.

Step 5: Approximate the Solution

On the graph, the lines intersect at the point \((3, 6)\). Check both equations to confirm: \(y = -2(3) + 12 = 6\) and \(y = 3 + 3 = 6\).Both equations are satisfied at this point.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that create straight lines when graphed on a coordinate plane. They are usually represented in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) the y-intercept. In the context of solving systems of equations, we often deal with two or more linear equations simultaneously.

The main goal when dealing with linear equations in systems is to find a solution that satisfies all equations at the same time. The solution to these systems is typically an ordered pair \((x, y)\) that lies on all the graphed lines at a common point. This is why understanding linear equations is crucial for solving them graphically.

When you have a good grasp of linear equations, you can easily identify if the system has one solution, no solutions, or infinitely many solutions. This depends entirely on how the lines related to the equations are positioned relative to each other.

Slope-Intercept Form

The slope-intercept form of a linear equation is \( y = mx + b \). This is one of the most common and easiest ways to represent a linear equation. It breaks down the equation into two main components: the slope \( (m) \) and the y-intercept \( (b) \).

• Slope (m): This indicates the steepness of the line and its direction. A positive slope means the line is inclined upwards, whereas a negative slope means it's going downwards.
• Y-intercept (b): This is where the line crosses the y-axis. It's the value of \( y \) when \( x \) is zero.

The slope-intercept form is particularly useful when graphing lines because it gives you straightforward instructions to plot. Start with the y-intercept, then use the slope to find another point on the line. This is done by using the rise over run principle, where you move vertically (rise) and horizontally (run) from the y-intercept, making it very simple and intuitive to work with.

Intersection Point

The intersection point of two lines is a critical concept when solving a system of equations graphically. It represents the \((x, y)\) values where both equations are true simultaneously. Finding this point graphically involves drawing both lines on a graph and identifying their crossing point.

In a system of linear equations, if the lines intersect at a single point, that point is the unique solution to the system. For our specific problem, the intersection point was found at \((3, 6)\). To ensure accuracy, this point can be substituted back into the original equations to verify that it satisfies both.

• Equation 1: \( y = -2x + 12 \)
• Equation 2: \( y = x + 3 \)

Substituting \( x = 3 \) and \( y = 6 \) into both equations, we find both hold true: - For Equation 1, \( y = -2(3) + 12 = 6 \)- For Equation 2, \( y = 3 + 3 = 6 \)

Thus, confirming the intersection point as the correct solution.

Graphing Techniques

When solving systems of equations graphically, using effective graphing techniques is crucial. Start by ensuring each equation is in slope-intercept form. This makes it easier to identify the y-intercept and the slope quickly, streamlining the plotting process.

To graph a line:

• First, plot the y-intercept on the graph. This is your starting point.
• Next, use the slope to find another point. Determine how many units to move up or down (rise) and how many to move right (run).
• Draw a straight line through the two points you've plotted.

The key element when graphing is precision, as the visual representation of the lines will determine the intersection point. Use a ruler or digital graphing tool if available to ensure straight lines.

While finding the intersection accurately might require zooming in or utilizing graphing software, practicing these techniques on paper helps develop a strong understanding of how changes in the equation affect the line's position and slope.