Problem 94 - Algebra 1: Concepts and Skills

Question

Graph the linear system and estimate a solution. Then check your solution algebraically. $$ \begin{aligned} &y=-3\\\ &x=4 \end{aligned} $$

Step-by-Step Solution

Answer

The system of equations intersects at the point (4,-3).

Step 1: Graph the first equation

Start by graphing the equation \(y=-3\). This is a simple horizontal line where \(y\) is always equal to -3, no matter what \(x\) is.

Step 2: Graph the second equation

Next, graph the second equation \(x=4\). This is also a simple line, but it is vertical where \(x\) is always 4, no matter what the value of \(y\) is.

Step 3: Find the intersection

Now, we need to find the intersection of these two lines. On the graph, you will see that the horizontal line \(y=-3\) and the vertical line \(x=4\) intersect at one point. That point is the solution to the system.

Step 4: Check the solution

To check the solution algebraically, substitute \(x=4\) into the first equation and \(y=-3\) into the second equation. It will satisfy both equations and confirm that the solution is correct.

Key Concepts

Graphing Techniques in Linear Systems

Graphing is a powerful technique used to solve linear systems of equations. Linear systems consist of two or more linear equations that you plot on the same set of axes. In our example, we have "y = -3" and "x = 4." Start by recognizing that these equations represent lines. Always remember:

• The equation "y = -3" indicates a horizontal line where every point has a y-coordinate of -3.
• The equation "x = 4" is a vertical line where every point has an x-coordinate of 4.

Plotting these on a graph involves drawing a straight line for each equation. The horizontal line passes through all points where y equals -3, and the vertical line passes through all points where x equals 4.
Paying close attention to these simple characteristics can help make graphing straightforward and effective.

Understanding Intersections

The intersection of two lines refers to the point where they cross each other on a graph. In the context of linear systems, finding the intersection is equivalent to finding the solution of the system.
For our equations "y = -3" and "x = 4," the graph shows a cross at a particular point. This is because:

• The horizontal line extends infinitely left and right at y = -3.
• The vertical line extends infinitely up and down at x = 4.

The intersection of these lines is at the point (4, -3). This single point represents the solution to our linear system, meaning it satisfies both equations simultaneously. Recognizing intersections visually in graphed systems is key to understanding solutions to linear equations.

Solution Checking in Linear Systems

Solution checking involves verifying that your intersection point indeed satisfies both equations in your system. It's an essential final step to ensure your solution is accurate.
Let's check our solution point (4, -3) algebraically:

• For the equation "y = -3," substitute y with -3. It clearly holds as \(-3 = -3\).
• For the equation "x = 4," substitute x with 4. Here, it also fits since \(4 = 4\).

Both substitutions satisfy their respective equations, which confirms that (4, -3) is indeed the correct solution.
Solution checking is crucial because it reinforces your understanding and ensures that your answer is indisputably correct. Remember, a correct solution satisfies all equations in the system.