Problem 21 - Algebra 1

Question

Graph and check to solve the linear system. $$ \begin{aligned} y &=-6 \\ x &=6 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is the point (6,-6).

Step 1: Graph the equation \(y = -6\)

Start by graphing the line \(y = -6\) which is a horizontal line across the y-axis at the point -6. This line has a slope of 0 and runs parallel to the x-axis.

Step 2: Graph the equation \(x = 6\)

Next, graph the line \(x = 6\) which is a vertical line on x-axis at the point 6. This line runs parallel to the y-axis.

Step 3: Find the intersection point

Now, determine the point where both of these lines intersect. This intersection point is the solution to the system of equations.

Step 4: Verify the solution

Finally, verify the solution by substituting \(x = 6\) and \(y = -6\) into both equations. If both equations are satisfied, then the solution is correct.

Key Concepts

Understanding System of Equations

A system of equations consists of two or more equations that share common variables. In our example, we have the equations \( y = -6 \) and \( x = 6 \). These equations form a system because they both share the variables \( x \) and \( y \).

The goal when working with systems of equations is to find values for the variables that satisfy all the equations at the same time. In other words, we're looking for a set of \( x \) and \( y \) values that work for both equations simultaneously.

Different methods exist for solving systems of equations, such as graphing, substitution, and elimination. Each approach has its own advantages, but graphing offers a visual representation that can make understanding the relationships between the equations easier, especially when dealing with simple equations.

Graphing to Solve Systems

Graphing is a method used to visually solve a system of equations, making it easier to see any relationships between the equations.

For the system \( y = -6 \) and \( x = 6 \), graphing involves drawing each equation on a coordinate plane.
- **Horizontal Line:** The equation \( y = -6 \) is graphed as a horizontal line across the plane at \( y = -6 \). This means it crosses the y-axis at -6 and runs parallel to the x-axis.
- **Vertical Line:** The equation \( x = 6 \) is graphed as a vertical line, which means it crosses the x-axis at 6 and runs parallel to the y-axis.

Graphing each line provides a straightforward way to analyze where they intersect, which ultimately gives us the solution to the system of equations.
Additionally, this method can be particularly effective for linear equations, providing a clear picture of how the equations relate to each other in the coordinate plane.

Identifying the Intersection Point

The intersection point of graphically plotted lines is a significant concept in solving systems of equations.

Once the lines for \( y = -6 \) and \( x = 6 \) are drawn, the point where they cross or meet is called the intersection point.
This specific point represents the values of \( x \) and \( y \) that satisfy both equations simultaneously.

In this problem, the intersection is at the point \( (6, -6) \). The x-coordinate is from the vertical line (\( x = 6 \)), and the y-coordinate comes from the horizontal line (\( y = -6 \)).

It's crucial to verify whether this point indeed solves the original system of equations. Substituting both coordinates into the equations should satisfy them:

• For \( y = -6 \), substituting \( y = -6 \) holds true.
• For \( x = 6 \), substituting \( x = 6 \) satisfies the equation.

If both conditions hold, which they do here, the intersection point is confirmed as the solution to the system.