Problem 14 - Algebra and Trigonometry

Question

\(13-14\) . Two equations and their graphs are given. Find the inter- section point(s) of the graphs by solving the system. $$ \left\\{\begin{array}{r}{x+y=2} \\ {2 x+y=5}\end{array}\right. $$

Step-by-Step Solution

Answer

The intersection point is (3, -1).

Step 1: Write the System of Equations Clearly

We are given the system of linear equations: \(x + y = 2\) and \(2x + y = 5\). Our goal is to find the intersection point(s) of the graphs of these equations by solving the system simultaneously.

Step 2: Solve for y in One Equation

Let's isolate \(y\) in the first equation: \(x + y = 2\). Subtract \(x\) from both sides to get \(y = 2 - x\).

Step 3: Substitute y in the Second Equation

Replace \(y\) in the second equation \(2x + y = 5\) with the expression we found: \(y = 2 - x\). This gives us \(2x + (2 - x) = 5\).

Step 4: Simplify and Solve for x

Simplify the equation: \(2x + 2 - x = 5\). Combine like terms to obtain \(x + 2 = 5\). Subtract 2 from both sides to find \(x = 3\).

Step 5: Solve for y Using x Value

Substitute \(x = 3\) back into the equation \(y = 2 - x\) to find \(y\). \(y = 2 - 3 = -1\).

Step 6: Verify the Solution

Check the point \((3, -1)\) in both original equations. For the first equation, \(3 + (-1) = 2\), which is true. For the second equation, \(2(3) + (-1) = 5\), which is also true. Both checks confirm the solution.

Key Concepts

Intersection Points

An intersection point is a point where two or more equations meet on a graph. Think of it as where the paths of the equations cross. For a system of linear equations, the intersection point is the solution that satisfies all equations at the same time. This means that if you plug in the values of this point into each equation, the equations hold true.

In our exercise, the intersection point is found by solving the equations simultaneously. Here, it was determined to be

• (3, -1) by solving both equations together.

This point represents a specific solution that tells us exactly where the two lines meet when plotted on a coordinate grid.

Graphing Equations

Graphing equations is a visual way to explore solutions to a system by plotting on a graph. When you graph a linear equation, it forms a straight line. The graph helps in identifying the intersection points easily, where the graphs (lines) of two equations meet.

For each equation in the system given:

• The equation \(x + y = 2\) forms one line on the graph.
• The equation \(2x + y = 5\) forms another line.

By plotting both lines on the same graph, we can observe the point at which they intersect. This intersection is the solution to the system, confirming that visually reviewing graphs provides context to the algebraic solution.

Solving Linear Systems

Solving linear systems involves finding a set of values that satisfy all equations in the system simultaneously. There are several methods to achieve this:

• Graphically, by examining the intersection point of lines.
• Algebraically, using substitution or elimination methods.

In the exercise, we used the substitution method. By isolating one variable in the first equation \(y = 2 - x\), we substituted it into the second equation and solved for \(x\).
Pursuing this path allowed us to solve for the variable \(y\), leading to the solution of the system. Always remember, solving linear systems will provide either a single solution (one point), infinitely many solutions (overlapping lines), or no solution (parallel lines).

Simultaneous Equations

Simultaneous equations are equations that are solved together, aiming to find a common solution. They are crucial for understanding situations where multiple conditions must be met at once. In our problem, we dealt with two simultaneous equations:

• \(x + y = 2\)
• \(2x + y = 5\)

These equations were solved at the same time to yield a distinct point that satisfies both conditions. Solving simultaneous equations often involves techniques such as substitution, like we did in the exercise, or elimination, which is another common method. The goal is always to find values for the variables that work for all equations, representing a feasible solution to the problem.