Problem 17 - Precalculus: Mathematical for Calculus

Question

\(17-22=\) Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system. $$\left\\{\begin{aligned}2 x+y &=-1 \\\x-2 y &=-8\end{aligned}\right.$$ GRAPH CANT COPY

Step-by-Step Solution

Answer

The intersection point of the graphs is \((-2, 3)\).

Step 1: Solve the First Equation for y

Start with the first equation: \(2x + y = -1\). Isolate \(y\) on one side by subtracting \(2x\) from both sides. We get: \[y = -2x - 1\]

Step 2: Substitute y in the Second Equation

Take the expression for \(y\) from Step 1 (\(y = -2x - 1\)) and substitute it into the second equation \(x - 2y = -8\). This gives us: \[x - 2(-2x - 1) = -8\] Simplify and solve for \(x\).

Step 3: Simplify and Solve for x

Distribute \(-2\) into \(-2x - 1\):\[x + 4x + 2 = -8\]Combine like terms: \[5x + 2 = -8\]. Subtract \(2\) from both sides: \[5x = -10\]. Finally, divide by \(5\): \[x = -2\].

Step 4: Substitute x Back to Find y

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

Step 5: Verify the Solution with Both Equations

Verify by plugging \(x = -2\) and \(y = 3\) into both original equations:1. For \(2x + y = -1\): \(2(-2) + 3 = -4 + 3 = -1\), which satisfies the first equation.2. For \(x - 2y = -8\):\(-2 - 2(3) = -2 - 6 = -8\), which satisfies the second equation.Since both equations are satisfied, the solution is correct.

Key Concepts

intersection points

When working with a system of equations, finding the intersection point is crucial. The intersection point is where two graphs cross one another on a coordinate plane.
In practice, this point gives the solution to both equations involved.
For the equations given:

• \(2x + y = -1\)
• \(x - 2y = -8\)

Solving them will give us a single point \((x, y)\) that solves both equations simultaneously.
The point \((-2, 3)\) was found to be this intersection point.
This means that if you plot both equations on the same graph, they will cross at \((-2, 3)\).
It confirms that both conditions set by the equations are held true at this specific point.Solving for intersection provides clarity and understanding, linking algebraic solutions with geometric meaning.

graph of equations

When you visualize the system of equations as a graph, you get a clearer picture of the problem.
A graph of a linear equation is a straight line, defined by all the \((x, y)\) pairs that solve the equation.
For such a system:

• The first equation, \(2x + y = -1\), can be rewritten in slope-intercept form as \(y = -2x - 1\).
• The second equation, \(x - 2y = -8\), can be transformed to plot by finding its intercepts or rewriting similarly.

When plotted on a graph, each line represents all possible solutions of the equation.
The coordinate plane becomes a tool where you can visually identify their common solution, or intersection.
The crossing point of these two lines represents the single solution to both equations.
This visual approach often helps in better understanding and verifying solutions derived from algebraic calculations.

substitution method

The substitution method is an effective technique to solve a system of equations.
It involves solving one equation for one variable and substituting that expression into the other equation.
This reduces two equations with two variables into a single equation with one variable.
For instance:

• We start with \(2x + y = -1\) and solve for \(y\), getting \(y = -2x - 1\).
• Substituting \(y\) into the second equation, \(x - 2y = -8\), simplifies the system and allows us to solve for \(x\).

Converting the system in such a manner simplifies the problem drastically.
Once \(x\) is found, pluggin it back into the expression for \(y\) gives its value.
The method boosts efficiency and precision in manual calculations, and also aids in solidifying your understanding of the relationship between the variables.