Problem 17 - Algebra and Trigonometry

Question

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system. \(\left\\{\begin{array}{c}{2 x+y=-1} \\ {x-2 y=-8}\end{array}\right.\)

Step-by-Step Solution

Answer

The graphs intersect at \((-2, 3)\).

Step 1: Understand the Equations

Here we have a system of linear equations: \(2x + y = -1\) and \(x - 2y = -8\). Our goal is to find the point where both these equations intersect, which is the solution of the system.

Step 2: Solve for y in one equation

Let's solve the first equation for \(y\). Starting with \(2x + y = -1\), we subtract \(2x\) from both sides, giving us: \(y = -2x - 1\).

Step 3: Substitute for y in the second equation

Now, take the expression for \(y\) from Step 2 and substitute it into the second equation \(x - 2y = -8\). This becomes \(x - 2(-2x - 1) = -8\).

Step 4: Simplify and solve for x

Simplify \(x - 2(-2x - 1) = -8\) to \(x + 4x + 2 = -8\), which simplifies further to \(5x + 2 = -8\). Subtract 2 from both sides to get \(5x = -10\), and then divide both sides by 5 to find \(x = -2\).

Step 5: Substitute x back to find y

Now substitute \(x = -2\) back into the expression for \(y\) from Step 2: \(y = -2(-2) - 1 = 4 - 1 = 3\). Thus, \(y = 3\).

Step 6: Verify the solution

Check the solution \((x, y) = (-2, 3)\) in the original equations. For the first equation, \(2(-2) + 3 = -4 + 3 = -1\) is correct. For the second equation, \(-2 - 2 \times 3 = -2 - 6 = -8\) is also correct.

Key Concepts

Intersection Points

In the world of geometry, an intersection point is where two lines meet or cross each other. When you're dealing with lines represented by equations, the intersection point is the set of coordinates (x, y) that satisfies both equations simultaneously.
Understanding intersection points allows you to find solutions for systems of equations, which is critical for solving numerous mathematical problems.

• The point of intersection tells you where two equations hold true together.
• It provides a concrete set of values that both equations share.

It's important to remember that an intersection point may not always exist if the lines are parallel. However, when they do exist, they are invaluable in determining the graphical solution of a system of equations.

Solving Equations

When we tackle a system of equations, our mission is to determine the values for variables that satisfy all equations. This process is known as solving equations.

For a system with two linear equations, one effective method is by substitution or elimination. In our case, we used substitution:

• First, solve one equation for one variable.
• Substitute that expression into the other equation.
• Solve for the remaining variable.
• Finally, substitute back to find the corresponding value of the initial variable.

This method focuses on rearranging equations to express one variable in terms of the other, leading to the solution that satisfies both original equations.

Linear Equations

Linear equations are equations that form straight lines when graphed on a coordinate plane. These equations typically involve constants and variables raised only to the first power.

In a system of linear equations, like the one we're examining, each equation forms a straight line:

• The equation \(2x + y = -1\) represents one line.
• The equation \(x - 2y = -8\) represents another line.

Each equation can be expressed in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.
Simplifying these equations and finding intersections helps solve real-world problems such as calculating distances, budgeting finances, or predicting trends.

Graphical Solutions

Graphical solutions involve drawing the equations on a coordinate plane and visually identifying where the lines intersect.
This method offers a more intuitive understanding of equations and their relationships:

• Plot each linear equation as a line on a graph.
• Identify the intersection point where the two lines meet.
• This point represents the solution to the system of equations.

Graphical solutions are particularly useful for visual learners and can provide a quick check against algebraic solutions.
They show if lines are parallel (no intersection), coincide (infinite intersections), or intersect once (one solution), highlighting relationships beyond numerical data.