Problem 1 - Basic Engineering Mathematics

Question

Solve graphically the simultaneous equations $$ \begin{array}{r} 2 x-y=4 \\ x+y=5 \end{array} $$

Step-by-Step Solution

Answer

The solution is the point (3, 2).

Step 1: Rewrite Equations for Graphing

First, we need to express each equation in the form of \(y = mx + b\), which is the standard linear format for graphing. For the first equation, \(2x - y = 4\), solve for \(y\):\[y = 2x - 4\].For the second equation, \(x + y = 5\), solve for \(y\):\[y = -x + 5\].

Step 2: Plot the First Line

Using the equation \(y = 2x - 4\), identify two points to draw the line.If \(x = 0\), then \(y = 2(0) - 4 = -4\). So, the point is \((0, -4)\).If \(x = 2\), then \(y = 2(2) - 4 = 0\). So, the point is \((2, 0)\).Plot these points and draw the line through them.

Step 3: Plot the Second Line

Using the equation \(y = -x + 5\), find two points.If \(x = 0\), then \(y = -0 + 5 = 5\). So, the point is \((0, 5)\).If \(x = 5\), then \(y = -5 + 5 = 0\). So, the point is \((5, 0)\).Plot these points and draw the line through them.

Step 4: Identify the Intersection Point

Look for the point where the two lines intersect on the graph.The intersection point is the solution to the system of equations.In this case, the lines intersect at the point \((3, 2)\).

Step 5: Verify the Solution Algebraically

Substitute \(x = 3\) and \(y = 2\) into both original equations to ensure they hold true.For the first equation, \(2(3) - 2 = 6 - 2 = 4\), which is correct.For the second equation, \(3 + 2 = 5\), which is also correct.Thus, the solution \((3, 2)\) is verified.

Key Concepts

Graphical Methods

Graphical methods offer a visual approach to solving simultaneous equations, which is often more intuitive for learners. By graphing the equations, you can visually identify where the two lines intersect, which represents the solution to the system.

To graph these equations, they must first be rewritten in the slope-intercept form, which is expressed as \(y = mx + b\). This format makes plotting straightforward because \(m\) is the slope and \(b\) is the y-intercept. Once the equations are graphed on the coordinate plane, the intersection point reveals the values of \(x\) and \(y\) that satisfy both equations simultaneously.

This method is particularly useful in classrooms for demonstrating solutions in an engaging, visual format, allowing students to grasp the concept of linear relationships quickly.

Linear Equations

Linear equations are mathematical statements establishing equality between two algebraic expressions, and they form straight lines when graphed. The key to their simplicity lies in their constant rate of change, represented by the slope.

In this exercise, the given equations are \(2x - y = 4\) and \(x + y = 5\). By transforming them into the standard slope-intercept form, we obtain \(y = 2x - 4\) and \(y = -x + 5\).

This transformation allows us to easily determine two distinct points for each equation needed to draw them on the graph. The fundamental principle here is that you can create a line with just two points, which is why finding these points is a critical step in solving simultaneous equations graphically.

Intersection Point

The concept of an intersection point is vital when solving systems graphically. It is the point where the lines representing the equations meet on the graph, signifying that the values of \(x\) and \(y\) satisfy both equations.

In this example, the lines based on \(y = 2x - 4\) and \(y = -x + 5\) intersect at the point \((3,2)\). This means that substituting \(x = 3\) and \(y = 2\) into the original equations will hold true for both.

Understanding the intersection point not only provides the solution but also offers insight into relationships in real-world applications, such as determining where two trends coincide. This concept underscores the importance of graphing in interpreting and understanding mathematical solutions.

Verification of Solutions

Verification of solutions ensures that the graphically determined intersection point is indeed the correct solution to the simultaneous equations. This is done by substituting the intersection point back into the original equations and confirming that both equations are satisfied.

For our example, substituting the values \((3,2)\) into the first equation \(2x - y = 4\) results in \(2(3) - 2 = 4\), which holds true. Similarly, inserting them into the second equation \(x + y = 5\) results in \(3 + 2 = 5\), confirming its accuracy.

This step is crucial as it validates the graphical solution and ensures the reliability of the results. It also reinforces the learning process by demonstrating that a correct graphical solution will always satisfy the original algebraic equations.