Problem 75 - Algebra 1

Question

GRAPHING LINEAR SYSTEMS Use the graphing method to solve the linear system and describe its solution(s). $$ \begin{aligned} &4 x+2 y=12\\\ &-6 x+3 y=6 \end{aligned} $$

Step-by-Step Solution

Answer

Graphically, the line \(y = -2x + 6\) and \(y = 2x + 2\) intersect at the point (1, 4). Therefore, the solution to the system of equations is the point (1, 4).

Step 1: Rewrite in Slope-Intercept Form

First, rewrite each equation in the y = mx + b form. For the first equation, \(4x + 2y = 12\), divide every term by 2. This gives \(2x + y = 6\), or \(y = -2x + 6\). For the second equation, \(-6x + 3y = 6\), divide every term by 3. This gives \(-2x + y = 2\), or \(y = 2x + 2\).

Step 2: Graph the Equations

The next step is to graph these equations. You could do it by choosing values for x and then solving for y. Make sure to choose at least two values for x so you can graph a line. The line for the equation \(y = -2x + 6\) will be downward sloping, and the line for \(y = 2x + 2\) will be upward sloping.

Step 3: Find the Point of Intersection

The solution to the system of equations is the point where the lines intersect. If the lines cross at a specific point, then that point is the unique solution. If the two lines are parallel (and therefore do not intersect), then there are no solutions. If the two lines coincide, then there are infinitely many solutions.

Key Concepts

Slope-Intercept Form

In algebra, the slope-intercept form is a way of writing a linear equation. It looks like this:

• \(y = mx + b\)

Here, \(m\) represents the slope of the line, while \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
To convert an equation to slope-intercept form, solve the equation for \(y\). Let's see how this works with the equation \(4x + 2y = 12\):
First, isolate the term with \(y\) by moving \(4x\) to the other side, which gives you \(2y = -4x + 12\). Next, divide each term by 2 to solve for \(y\):
\(y = -2x + 6\).
Now your equation is in slope-intercept form, and you can easily see the slope \(-2\) and the y-intercept \(6\).
This form is very helpful for graphing because it clearly shows how steep the line is and where it starts on the graph.

System of Equations

A system of equations consists of two or more equations with the same set of unknowns. In our example, we have:

• \(4x + 2y = 12\)
• \(-6x + 3y = 6\)

Solving a system of equations means finding the values of the variables that satisfy all equations at the same time.
There are several methods to solve systems of equations: substitution, elimination, and graphing.
In this problem, we're using the graphing method, which involves drawing each equation on a graph and looking for the intersection point.
The solution to the system is the point \((x, y)\) where the graphs of the equations cross each other.

Point of Intersection

The point of intersection is crucial when solving a system of equations graphically. This point is where the lines representing the equations meet on a graph.
The coordinates of this point are the values of the variables that solve the system. For instance, if two lines intersect at \((3, 4)\), that means \(x = 3\) and \(y = 4\) satisfy both equations.
There are different outcomes when graphing two lines:

• If they intersect at a single point, there's a unique solution.
• If they are parallel, they never meet, meaning no solution.
• If they coincide (are the same line), there are infinitely many solutions.

In our problem, finding this point confirms whether a unique solution exists or not.

Graphing Method

The graphing method is one of the simplest ways to solve a system of equations, especially when you want a visual understanding.
Here’s a step-by-step on how it works:

• Rewrite each equation in slope-intercept form \(y = mx + b\).
• Draw the graph of each line using the slope \(m\) and y-intercept \(b\).
• Observe where the lines intersect on the graph.

This method offers a clear picture of the relationships between the equations.
However, it’s sometimes not very precise if the point of intersection does not fall exactly on grid lines. This is why it works best for systems with simple solutions or for getting an idea of the nature of the solutions.
Always double-check with another method if you need precise answers.