Problem 85 - College Algebra Essentials

Question

Graph \(x+2 y=2\) and \(x-2 y=6\) in the same rectangular coordinate system. At what point do the graphs intersect? $$ \text { Solve: } 5(2 x-3)-4 x=9 $$

Step-by-Step Solution

Answer

The two given lines intersect at the point (2, -1). The solution to the equation \(5(2x - 3) - 4x = 9\) is \(x = 4\).

Step 1: Convert Equations to Slope-Intercept Form

To graph the lines, let's first get the equations in the form of \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. This can be done by rearranging each equation to isolate \(y\). Our two equations are: \(x+2y=2\) and \(x-2y=6\), which become \(y = -0.5x + 1\) and \(y = 0.5x - 3\).

Step 2: Graph the Equations

For each equation, plot the y-intercept (where the line crosses the y-axis), then use the slope as a set of directions from the y-intercept to locate another point on each line. Draw these two lines out on the same grid. For \(y = -0.5x + 1\), plot (\(0, 1\)) and move down 0.5 units and right one unit to plot another point, draw the line. For \(y = 0.5x - 3\), plot (\(0, -3\)) and move up 0.5 units and right one unit to plot another point, draw the line.

Step 3: Identify Intersection Point

The intersection point of the two lines is where the solution to the system exists. Looking at the plotted lines we can find that they intersect at the point (\(2, -1\)).

Step 4: Simplify Complex Equation

Now to solve the single equation \(5(2x - 3) - 4x = 9\), begin by distributing the \(5\) inside the parentheses, then combine like terms on the left side. This results in \(10x - 15 - 4x = 9\), then simplify to \(6x - 15 = 9\).

Step 5: Solve for 'x'

Finally, solve for 'x' by first adding \(15\) to both sides, then divide by \(6\) to isolate 'x'. So, \(x = \frac{24}{6} = 4\).

Key Concepts

Slope-Intercept Form

Understanding slope-intercept form is crucial when graphing linear equations. The slope-intercept form of a line is given by the equation \(y = mx + c\). Here, \(m\) represents the slope, which indicates the steepness or inclination of the line, and \(c\) represents the y-intercept, the point where the line crosses the y-axis.

To convert a linear equation into slope-intercept form, you need to isolate \(y\) on one side of the equation. This usually involves rearranging the terms and sometimes performing basic operations like addition, subtraction, multiplication, or division.

For example, for the equation \(x + 2y = 2\), you can solve for \(y\) by first subtracting \(x\) from both sides, getting \(2y = -x + 2\), and then dividing each term by 2 to get \(y = -0.5x + 1\). Similarly, the equation \(x - 2y = 6\) can be converted to \(y = 0.5x - 3\).

This form helps quickly identify both the slope and the y-intercept, which are essential for graphing the line on a coordinate plane.

System of Equations

A system of equations consists of two or more equations that share the same set of variables. The core idea is to find a common solution or solutions that satisfy all equations in the system simultaneously. In our case, the system is represented by two linear equations: \(x + 2y = 2\) and \(x - 2y = 6\).

The solutions to these systems can be found by graphing the equations on the same set of axes and identifying points where the lines intersect. There are also algebraic methods to find the solutions, such as substitution or elimination, but graphing provides a visual understanding.

In practice, solving a system of equations can involve:

• Writing each equation in slope-intercept form to facilitate easy graphing.
• Plotting each line accurately on the graph.
• Identifying the intersection points of the lines, which are the solutions to the system.

This process gives a tangible insight into how different equations relate in a geometric sense.

Intersection of Lines

The intersection of lines is the point where two lines cross each other on a graph. This point is significant because it represents the solution to the system of equations.

When graphing, once the lines are plotted based on their slope and y-intercept, the intersection can usually be visually identified. In our scenario, the lines described by \(y = -0.5x + 1\) and \(y = 0.5x - 3\) intersect at the point \((2, -1)\). This point satisfies both equations simultaneously, making it the solution to the system.

Finding the intersection point involves ensuring the lines are correctly plotted by:

• Checking that the slopes are correctly interpreted.
• Ensuring that the y-intercepts are plotted accurately.
• Carefully drawing the lines based on multiple points if necessary.

If plotted accurately, the intersection reveals the point of balance or agreement between the equations, and this is the solution we seek.

Solving Linear Equations

Solving linear equations involves manipulating an equation to find the unknown variable. Linear equations are typically represented in the form \(ax + b = c\), where the solutions can be found using algebraic operations.

The provided equation, \(5(2x - 3) - 4x = 9\), involves several steps:

• First, distribute the 5 inside the parentheses: \(10x - 15\).
• Then, combine like terms: \(10x - 4x - 15 = 9\) simplifies to \(6x - 15 = 9\).
• Next, isolate the \(x\) by adding 15 to both sides, resulting in \(6x = 24\).
• Finally, divide each side by 6 to solve for \(x\): \(x = 4\).

Each equation requires a systematic approach to isolate the variable of interest. The final answer often provides a basis for further calculations or helps understand the context of the problem.