Problem 8 - Thinking Mathematically

Question

Solve each system by graphing. Check the coordinates of the intersection point in both equations. \(\left\\{\begin{array}{l}4 x+y=4 \\ 3 x-y=3\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system of equations is the point (1, 0).

Step 1: Rearrange the equations

To rearrange the first equation to isolate y, start with \(4x + y = 4\). Subtract \(4x\) from both sides to get \(y = -4x + 4\). For the second equation, start with \(3x - y = 3\). Add \(y\) and subtract \(3\) from both sides to get \(y = 3x - 3\).

Step 2: Plot the lines

Graph the two equations on a coordinate plane. You can see that they intersect at the point \(1, 0\).

Step 3: Check the solution

Substitute the values of \(x = 1\) and \(y = 0\) into the original equations to make sure they are indeed correct. Substituting into \(4x + y = 4\) gives \(4*1 + 0 = 4\) which holds true. Substituting into \(3x - y = 3\) gives \(3*1 - 0 = 3\), which also checks out.

Key Concepts

Intersection Point

When graphing a system of equations, the intersection point is the solution to the system. This is where the graphs of the equations meet on the coordinate plane. Understanding how to find and verify the intersection point is crucial in solving equations by graphing.

In the exercise provided, you find the intersection point by plotting the two linear equations. The point where the two lines cross their paths is the solution to our system. This is because at the intersection point, both equations are satisfied simultaneously. For instance, in our example, the intersection point is \(x = 1, y = 0\). This means both equations hold true when \(x\) is 1 and \(y\) is 0.

It is always important to "check" this point by substituting the coordinates back into the original equations to ensure they balance out correctly. Failure to do so could mean incorrect results.

Coordinate Plane

A coordinate plane is a crucial tool in graphing equations and understanding their relationships. It consists of two axes: the horizontal is the x-axis, and the vertical is the y-axis. Points on the plane are identified by their coordinates \(x, y\), where \(x\) represents the horizontal position, and \(y\) represents the vertical position.

Graphing on a coordinate plane allows us to visually interpret equations and see how different lines relate to each other. In our exercise, by graphing the lines \(y = -4x + 4\) and \(y = 3x - 3\), we could easily visualize how they intersect at \(1, 0\) on the plane. This visualization is helpful because it shows directly how the two equations converge at the solution point.

Understanding the layout and use of a coordinate plane can significantly enhance one's ability to solve mathematical problems. It provides a visual means of exploring algebraic concepts and solutions.

System of Equations

A system of equations consists of two or more equations with the same set of variables. Solving a system involves finding the values of the variables that satisfy all equations in the system simultaneously. There are several methods to solve systems of equations, such as graphing, substitution, or elimination. In this case, we focus on the graphing method.

When dealing with systems of equations like the one in our exercise, you start by graphing each equation on the coordinate plane. This will help you see where the lines intersect, thus providing the solution. The two equations provided, \(4x + y = 4\) and \(3x - y = 3\), interact in a way that they have exactly one intersection point. This means there is one solution that works for both equations.

Graphing is particularly useful for smaller systems or when you need a visual representation. However, it's important to consider other methods for larger or more complex systems that might not graph as neatly.