Problem 24 - Algebra A Combined Function

Question

Solve each system of linear equations by graphing. See Examples 3 through \(6 .\) \(\left\\{\begin{array}{l}x+y=5 \\ x=4\end{array}\right.\)

Step-by-Step Solution

Answer

The solution is the point \((4, 1)\).

Step 1: Understand the System of Equations

We have two equations: \(x + y = 5\) and \(x = 4\). Our task is to find the point(s) where these lines intersect by graphically representing them.

Step 2: Graph the First Equation

Let's graph the equation \(x + y = 5\). This is a line in the form \(y = -x + 5\). To graph it:- Start by finding two points: when \(x = 0\), then \(y = 5\) (point \((0,5)\)), and when \(y = 0\), then \(x = 5\) (point \((5,0)\)).- Plot these points on a graph and draw the line passing through them.

Step 3: Graph the Second Equation

The second equation is \(x = 4\). This is a vertical line that passes through all points where \(x = 4\).- Simply draw a straight vertical line that crosses the x-axis at \(x=4\).

Step 4: Find the Intersection Point

To solve the system, find the intersection point of these two lines on the graph.- The line \(x = 4\) will intersect the line \(x + y = 5\) at a single point.- Substitute \(x = 4\) into the first equation: \(4 + y = 5\), which simplifies to \(y = 1\).- Thus, the intersection point is \((4,1)\).

Step 5: Verify the Solution

Verify the solution point \((4,1)\) by plugging it back into both original equations:- For \(x + y = 5\): \(4 + 1 = 5\) which is true.- For \(x = 4\): \(x = 4\) which is also true. - Therefore, the solution is verified.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental skill in understanding and solving systems of equations. A linear equation, like those given in this exercise, typically takes the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. To graph an equation such as \( x + y = 5 \), it's often helpful to rearrange it into this form.

• Start by finding easy-to-calculate points, such as the x-intercept and y-intercept. In this equation, when \( x = 0 \), \( y \) is 5, thus giving the point \((0, 5)\).
• Similarly, when \( y = 0 \), \( x \) is 5, resulting in the point \((5, 0)\).
• Plot these points on a graph and connect them with a straight line.

This line represents all possible solutions to the equation \( x + y = 5 \). The second equation, \( x = 4 \), is simpler as it represents a vertical line crossing the x-axis at 4. This can be quickly graphed by drawing a line parallel to the y-axis through \( x = 4 \). By practicing graphing these lines, you gain a visual understanding of how these equations behave and where they intersect.

Intersection Point

An intersection point is the specific location where two lines meet on a graph. Identifying this point is crucial for solving systems of equations graphically. In this exercise, each equation represents a line, and the solution is where these lines intersect.For our system:

• Graph the line \( x + y = 5 \) by identifying and plotting two points, such as \((0, 5)\) and \((5, 0)\).
• Then, graph the vertical line \( x = 4 \), which intersects the \( x \)-axis at \(4\).
• The intersection of these two lines can visually be found at the point where they cross on the graph. In our case, this is point \((4, 1)\).

This intersection point satisfies both equations in the system, and hence, it is the solution to the system of equations. Understanding how to locate and interpret intersection points helps in determining solutions quickly without needing complex calculations.

Verifying Solutions

Once the intersection point is found, it's important to verify it to ensure that it satisfies all equations involved in the system. Verification is a critical step that confirms whether the solution is correct and consistent with each equation. To verify, you substitute the intersection point back into each original equation:

• First, for the line \( x + y = 5 \), substitute \( x = 4 \) and \( y = 1 \):

\[ 4 + 1 = 5 \]

• This equation holds true.
• Next, check for the line \( x = 4 \):

Simply note that \( x = 4 \), which is correct by the equation's own definition. Each equation validates the intersection point \((4, 1)\), confirming it as a legitimate solution to the system. Doing these checks reassures that no error was made either in graphing or calculation.

Linear Equation Systems

Linear equation systems consist of two or more equations that we solve simultaneously. They often represent real-life scenarios where multiple conditions apply at once. In this exercise, the system:

• Comprised of \( x + y = 5 \) and \( x = 4 \), illustrates two simple yet intersecting conditions.

To solve such systems graphically:

• Each equation is graphed and the intersection, if one exists, represents the solution.
• The system is classified based on outcomes: consistent systems have a unique solution, while inconsistent ones do not intersect and have no solution.
• Dependent systems have infinite solutions because the lines coincide.

Solving systems of linear equations provides insights into how different constraints relate in a shared space, making these skills essential for deeper mathematical understanding and practical problem-solving.