Problem 41 - College Algebra and Calculus: An Applied Approach

Question

Solve the system graphically. $$\left\\{\begin{array}{r}x-3 y=-2 \\ 5 x+3 y=17\end{array}\right.$$

Step-by-Step Solution

Answer

On graphing the two equations, it is found that they intersect at the point (3, 1). So, the solution to the system of equations is x = 3 and y = 1.

Step 1: Identify the Equations

The given system of equations is \[\begin{array}{r}x-3y=-2 \5x+3y=17\end{array}\]The aim is to find the point that satisfies both equations, which could be done graphically.

Step 2: Convert Equations to y = mx+b Form

First, the equations should be transformed to the slope-intercept form (y = mx+b). The first equation \(x - 3y = -2\) becomes \(y = \frac{1}{3}x + \frac{2}{3}\).The second equation \(5x + 3y = 17\) becomes \(y = -\frac{5}{3}x + \frac{17}{3}\).

Step 3: Graph the Two Equations

Plot the two equations \(y = \frac{1}{3}x + \frac{2}{3}\) and \(y = -\frac{5}{3}x + \frac{17}{3}\) on the same set of axes. The intersection point represents the solution to the system.

Step 4: Identify the Intersection Point

Observe the point where the two lines intersect. This point, which satisfies both equations, is the solution to the system of equations.

Key Concepts

Slope-intercept form

The slope-intercept form of a linear equation is written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) represents the y-intercept, the point where the line crosses the y-axis. Converting an equation to this form helps us quickly identify the slope and the y-intercept, which are vital for graphing the equation.

• The slope, \( m \), indicates the steepness of the line and the direction it moves. A positive slope means the line rises as you move from left to right, whereas a negative slope means it falls.
• The y-intercept, \( b \), is where the line crosses the y-axis. It is the starting point of the line when \( x = 0 \).

In the exercise, the first equation \( x - 3y = -2 \) was rearranged into \( y = \frac{1}{3}x + \frac{2}{3} \), indicating a slope of \( \frac{1}{3} \) and a y-intercept of \( \frac{2}{3} \). The second equation, \( 5x + 3y = 17 \), becomes \( y = -\frac{5}{3}x + \frac{17}{3} \), with a slope of \(-\frac{5}{3}\) and a y-intercept of \(\frac{17}{3}\).

Intersection point

The intersection point of two lines on a graph is where both lines cross each other. In the context of solving systems of equations graphically, this point represents the solution to the system. It is the set of coordinates \((x, y)\) that satisfies both equations simultaneously.

• To find the intersection point, graph both equations on the same set of axes.
• Look for the point where the lines meet.

This intersection provides a visual representation of the solution. For the given system of equations, the intersection offers the precise \(x\) and \(y\) values that are true for both equations at once.
The process involves sketching each line derived from the slope-intercept form of the equations and noting where they intersect. This point of intersection is crucial because it offers a tangible way to understand and verify the unique solution visually.

Linear equations

Linear equations are equations of the first degree, meaning they involve only the first power of the variable. They are called "linear" because their graph forms a straight line. This simplicity allows linear equations to serve as the foundation for solving complex systems graphically. Key characteristics of linear equations include:

• They can be written in several forms, such as slope-intercept \( y = mx + b \), standard form \( Ax + By = C \), and point-slope form \( y - y_{1} = m(x - x_{1}) \).
• Each form provides different insights into the properties of the linear equation and can be used to graph it conveniently on a coordinate plane.

In a system of equations like the one presented, each equation represents a line, and the solution is where these lines intersect. Understanding how to convert equations into usable forms and interpret their graphical representation is crucial for solving these systems and underscores the simplicity and power of linear equations in algebra.