Problem 77 - Introductory Algebra for College Students

Question

Will help you prepare for the material covered in the first section of the next chapter. Determine the point of intersection of the graphs of \(2 x+3 y=6\) and \(2 x+y=-2\) by graphing both equations in the same rectangular coordinate system.

Step-by-Step Solution

Answer

The point of intersection of the graphs of the given equations is (2, 0).

Step 1: Rearrange the equations

Rearrange both equations into slope-intercept form \(y = mx + c\). For \(2x + 3y = 6\), subtract \(2x\) from both sides to get \(3y = -2x + 6\), then divide all terms by 3 to isolate \(y\), giving \(y = -2/3x + 2\). Similarly for \(2x + y = -2\), subtract \(2x\) from both sides to get \(y = -2x - 2\).

Step 2: Plot the Lines

Now plot these two lines on the same graph. Plot the line \(y = -2/3x + 2\), which has a slope of -2/3 and a y-intercept of 2. Then plot the line \(y = -2x - 2\), which has a slope of -2 and a y-intercept of -2.

Step 3: Identify Intersection Point

The intersection of these lines is the solution to the simultaneous equations - it's the point that satisfies both equations. In this case, by looking at the graph, it can be observed that they intersect at the point (2, 0).

Key Concepts

Slope-Intercept Form

One of the most fundamental ways to express a linear equation is in slope-intercept form, which is denoted as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept. This form is very useful because it quickly communicates two key attributes of the line: its steepness and where it crosses the y-axis.
In the exercise provided, converting the given equations into slope-intercept form allows students to easily graph them. For example, by rewriting \( 2x + 3y = 6 \) into \( y = -\frac{2}{3}x + 2 \), it becomes clear that the line will cross the y-axis at 2 and for each step right (in the positive direction of the x-axis), it will move \( \frac{2}{3} \) units down. Similarly, \( y = -2x - 2 \) predicts a downward line with a steeper gradient, crossing the y-axis below the origin at -2.

Simultaneous Equations

Simultaneous equations are a set of equations with multiple unknowns that are true at the same time. To find the common solution, which is typically a point in the coordinate system where these equations intersect, various methods can be employed, such as substitution, elimination, or by graphing. In the exercise, we are using the graphing method.
Once the equations are in slope-intercept form, graphing them will visually demonstrate where they intersect. The intersection point corresponds to the values of the variables that satisfy both equations simultaneously. Therefore, finding this point is crucial in understanding how simultaneous equations can have a unique solution.

Coordinate System

Understanding the coordinate system is essential when graphing linear equations. The coordinate system, also known as the 'Cartesian coordinate system', consists of two axes that intersect at a right angle. The horizontal axis is referred to as the 'x-axis', while the vertical axis is called the 'y-axis'. Each point in this system is described by an ordered pair of numbers, where the first number represents the position along the x-axis and the second number along the y-axis.
When plotting the graph of a linear equation like those in the exercise, the slope-intercept form gives a starting point (the y-intercept) and the slope tells us how to move from that starting point to draw the line.

Intersection Point

The intersection point is where two or more lines on a graph cross each other, and it holds significant importance when solving simultaneous equations graphically. It represents the set of values that are solutions to all equations involved.
In the given exercise, after plotting the two lines, we note that they intersect at a specific point on the graph. This is the point (2, 0), meaning that when \( x=2 \), both equations yield the same value of \( y \), which in this case is zero. This shared point verifies that both original equations are satisfied for these specific values of \( x \) and \( y \).