Problem 30 - Elementary Algebra Within Reach

Question

In Exercises 27-36, solve the system by graphing. $$ \left\\{\begin{array}{l} 3 x+2 y=-6 \\ 3 x-2 y=6 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is the point (0, -3).

Step 1: Transform Equations into Slope-Intercept Form

The slope-intercept form of an equation is \( y = mx + c \), where m is the slope and c is the y-intercept. So, we need to transform the given equations into this form: For the first equation, \(3x + 2y = -6 \), we subtract \(3x\) from both sides to obtain \(2y = -3x -6 \). Divide each side by 2 and get the first equation in slope-intercept form: \(y = - \frac{3}{2}x - 3 \).For the second equation, \(3x - 2y = 6\),we subtract \(3x\) from both sides to get \(-2y = -3x + 6\). Dividing both sides by -2 to get\(y = \frac{3}{2}x - 3\).

Step 2: Draw the Graphs

Now, we can graph the lines \(y = -\frac{3}{2}x - 3\) and \(y =\frac{3}{2}x - 3\) on a set of coordinate axes. To do so, choose several values for \(x\), substitute them into the equations, find corresponding \(y\) values, plot those points, and draw the lines through them.

Step 3: Find Intersection Point

The solution to this system of equations is the coordinates of the point where these two lines intersect. From the graph, it can be seen that the lines cross at the point (0, -3). This gives us the solution to the system of equations.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of expressing linear equations with an emphasis on the slope, the tilt of the line, and the y-intercept, which is where the line crosses the y-axis. This form is written as:

• \( y = mx + c \)

Here, \( m \) represents the slope, and \( c \) denotes the y-intercept. This form is particularly helpful because it makes it easy to visualize how a line will look on a graph just by knowing these two components. The slope \( m \) indicates how steep the line is and in which direction it tilts. A positive slope tilts upwards, while a negative slope tilts downwards. The y-intercept \( c \) shows where the line crosses the y-axis, giving a starting point to draw the line. By rearranging a linear equation into the slope-intercept form, we make it more straightforward to graph the equation and analyze its properties.

Graphing Linear Equations

Graphing linear equations involves plotting their solutions on a coordinate plane to visualize them as lines. The slope-intercept form, \( y = mx + c \), provides an efficient mechanism for doing this. Here's a simple process to follow:

• Identify the slope \( m \) and the y-intercept \( c \) from the equation.
• Start by plotting the y-intercept on the y-axis. This is your first point.
• Use the slope to determine your next point. The slope is a ratio, \( \frac{rise}{run} \). For example, if \( m = \frac{3}{2} \), go up 3 units and run 2 units to the right to find the next point.
• Connect these points using a straight line.

Repeat these steps for each equation involved. This way, you create visual representations of the equations, aiding in the understanding of their relationships and intersections.

Intersection of Lines

The intersection of lines on a graph shows the solution to a system of equations. When you graph two linear equations, you are essentially looking to see if and where the lines cross each other. The point of intersection is the set of coordinates that satisfies both equations simultaneously.

• If the lines intersect at a single point, that point provides the solution to the system, giving specific values for \( x \) and \( y \).
• If the lines are parallel and never intersect, the system has no solution.
• If the lines are coincident, lying on top of each other, the system has infinitely many solutions.

In our example, the intersection occurs at the point (0, -3), meaning both equations equal each other precisely at this coordinate. This interplay is vital in understanding how different systems of equations can interact in two-dimensional space.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics allows relationships between geometric figures to be expressed algebraically.

• A standard coordinate system consists of two axes: the x-axis (horizontal) and the y-axis (vertical).
• Points in this system are denoted as coordinates, \( (x, y) \), representing their position relative to the origin (0,0).

By using equations, you can describe lines, curves, and other geometric figures in the coordinate plane. Coordinate geometry connects algebraic equations to geometric figures, thus providing tools to solve problems involving distances, midpoints, areas, and intersections such as the ones seen in solving systems of equations. The graph of a linear equation, such as those in our given system, represents a straight line, making it easier to comprehend how these mathematical entities interact visually.