Problem 13 - Precalculus Mathematics for Calculus

Question

Two equations and their graphs are given. Find the inter- section point(s) of the graphs by solving the system. $$\left\\{\begin{array}{r}2 x+y=-1 \\\x-2 y=-8\end{array}\right.$$ CAN'T COPY THE GRAPH

Step-by-Step Solution

Answer

The intersection point is (-2, 3).

Step 1: Setup the Equations

To find the intersection point of the two lines, we start by considering the system of equations:\[\begin{align*}2x + y &= -1 \x - 2y &= -8\end{align*}\]

Step 2: Solve for One Variable

Choose one of the equations to solve for one of the variables. Let's solve the second equation for \(x\):\[x - 2y = -8\]Rearrange to get:\[x = 2y - 8\]

Step 3: Substitute into the Other Equation

Substitute \(x = 2y - 8\) into the first equation:\[2(2y - 8) + y = -1\]

Step 4: Solve for y

Expand and solve the equation:\[4y - 16 + y = -1\]Combine like terms:\[5y - 16 = -1\]Add 16 to both sides:\[5y = 15\]Divide by 5:\[y = 3\]

Step 5: Substitute Back to Find x

Substitute \(y = 3\) back into \(x = 2y - 8\):\[x = 2(3) - 8\]Calculate to find:\[x = 6 - 8 = -2\]

Step 6: Write the Solution as an Ordered Pair

The solution to the system, which is the intersection point of the graphs, is \((-2, 3)\).

Key Concepts

Solving for Variables

When working with systems of equations, you often need to solve for one variable first. This means isolating a variable in one of the equations so you can use its expression in another equation. Let's break this down a bit.
Imagine you have two equations that share the same variables, like our example:

• Equation 1: \(2x + y = -1\)
• Equation 2: \(x - 2y = -8\)

To effectively solve this system, you decide to solve the second equation for \(x\). This involves rearranging the equation to express \(x\) in terms of \(y\), resulting in \(x = 2y - 8\). By doing this, you have isolated \(x\) on one side of the equation. Now, you have an expression for \(x\) that you can substitute into the first equation.
Substitution is a powerful tool for solving systems of equations and helps you simplify complex problems step by step.

Intersection Points

The key goal in solving systems of equations is to find intersection points. An intersection point is where two lines on a graph meet, representing a solution to the system.
In our example:

• After solving for \(y\) using substitution and simplification, you find \(y = 3\).
• You then substitute \(y = 3\) back into the expression for \(x\), giving you \(x = -2\).

These values \((-2, 3)\) form the coordinates of the intersection point. This means that if you were to graph both equations, the lines would cross at this exact point.
Finding intersection points is crucial because it shows where solutions exist, which can be incredibly useful in real-world problem-solving scenarios like determining market equilibrium or meeting point on a map.

Graphing Equations

Graphing equations can make understanding systems of equations much easier. Visualizing these equations as lines on a graph helps you see where they intersect.
To graph the given equations:

• For \(2x + y = -1\), rewrite it in slope-intercept form: \(y = -2x - 1\). This lets us quickly plot the y-intercept (-1) and use the slope (-2) to determine other points.
• For \(x - 2y = -8\), rearrange to \(y = \frac{1}{2}x + 4\). The y-intercept is 4, and the slope is \(\frac{1}{2}\).

By plotting each of these lines, you'll visualize where they cross, confirming your previous algebraic solution of \((-2, 3)\) as the intersection point.
Graphing is a great way to double-check your work and provide a visual representation of where solutions to systems of equations occur. It makes the process of algebra fun and more exciting by adding a visual dimension.