Problem 15 - Introductory Algebra for College Students

Question

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}2 x-3 y=6 \\ 4 x+3 y=12\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \{(3,0)\}.

Step 1: Rewrite The Equations In Slope-Intercept Form

The first equation \(2x - 3y = 6\) can be rewritten as \(y = \frac{2}{3}x - 2\). The second equation \(4x + 3y = 12\) can be rewritten as \(y = -\frac{4}{3}x + 4\).

Step 2: Draw The Graphs

Draw the lines represented by both equations on the same coordinate plane for better visualization. Don't forget to label each line with its equation.

Step 3: Find The Intersection Point

The point of intersection is the solution to the system of equations. By observing the graph, we see that the lines intersect at \(x = 3\), \(y = 0\) which is the solution.

Step 4: Write The Solution In Set Notation

Express the solution using set notation as \{(3,0)\}.

Key Concepts

Graphing Method

The graphing method is a visual way to solve systems of linear equations. It involves plotting each equation on a coordinate plane. Once both lines are graphed, the point where they intersect represents the solution to the system of equations. This method can be quite beneficial for visual learners who grasp concepts better through images than algebraic manipulation. Key steps to follow:

• Convert each equation into the slope-intercept form if needed.
• Graph each equation on the same coordinate grid.
• Identify the point where the lines intersect.

The graphing method not only helps in finding the solution but also gives a visual understanding of how solutions relate to the lines of the equations.

Slope-Intercept Form

The slope-intercept form of a linear equation is given by the formula:\[y = mx + b\]where \(m\) represents the slope of the line, and \(b\) denotes the y-intercept, which is the point where the line crosses the y-axis. This form makes it straightforward to graph an equation as it clearly indicates the steepness of the line and its starting position on the y-axis. In our example, the equations were:

• \(2x - 3y = 6\), converted to \(y = \frac{2}{3}x - 2\)
• \(4x + 3y = 12\), converted to \(y = -\frac{4}{3}x + 4\)

Notice how the coefficients of \(x\) change when isolating \(y\), reflecting the slopes of the respective lines. Understanding slope-intercept form allows you to quickly sketch the lines without extensive calculations.

Intersection Point

The intersection point in a graph of two lines is fundamentally where the two lines cross or meet. This point signifies the solution to the system of equations if one exists. For a solution to exist, the lines must intersect exactly once in the coordinate plane. If the lines are parallel and do not meet, the system has no solution. If the lines are identical, they intersect at every point along the line, creating infinitely many solutions. In our problem:- By graphing, we found the lines intersect at the point \((3,0)\).- This intersection point satisfies both equations simultaneously, confirming it as the correct solution.Understanding the intersection point helps to determine not only if a solution exists but what that solution is in terms of x and y values.

Set Notation

Set notation is a mathematical way to describe a collection of objects or numbers satisfying certain conditions. In the context of solving systems of equations, set notation helps to clearly express the solution of the system. For a single solution, as in this example, the set notation is written as:\[\{(3,0)\}\]This notation explicitly lists the solution point in the format \((x, y)\), where both x and y values satisfy the equations. Set notation can vary slightly depending on whether there is one solution, no solution, or infinitely many solutions:

• One Solution: \(\{(x,y)\}\)
• No Solution: \(\emptyset\)
• Infinite Solutions: Expressed generally, e.g., \(\{(x, f(x)) \, | \,x \in \mathbb{R}\}\)

Using set notation gives a precise and concise way to communicate solutions in mathematical terms.