Problem 13 - 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. $$ \left\\{\begin{aligned} 2 x+y &=-1 \\ x-2 y &=-8 \end{aligned}\right. $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

To solve the system of equations given, we need to find the point(s) where the two lines intersect on a graph. This requires us to find the values of \(x\) and \(y\) that satisfy both equations.

Step 2: Equations

We are given the two equations: \(2x + y = -1\) (Equation 1) and \(x - 2y = -8\) (Equation 2). Our task is to find the values of \(x\) and \(y\) that make both equations true.

Step 3: Substitution Method Setup

Let's solve Equation 1 for \(y\). Rearrange Equation 1: \( y = -1 - 2x \). Use this expression for \(y\) in Equation 2.

Step 4: Substitute \(y\) in Second Equation

Substitute \(y = -1 - 2x\) from Equation 1 into Equation 2: \(x - 2(-1 - 2x) = -8\). Simplify the equation to solve for \(x\).

Step 5: Simplifying and Solving for \(x\)

Simplify: \( x + 2 + 4x = -8 \). Combine like terms to get: \( 5x + 2 = -8 \). Subtract 2 from both sides: \( 5x = -10 \). Divide by 5 to solve for \(x\): \( x = -2 \).

Step 6: Substitute \(x\) back to find \(y\)

Substitute \(x = -2\) back into the expression for \(y\): \( y = -1 - 2(-2) \). Simplify this to find \(y\): \( y = -1 + 4 = 3 \).

Step 7: Verify the Solution

Now we have \(x = -2\) and \(y = 3\). Substitute these values into both original equations to ensure they make the equations true. When \(x = -2\) and \(y = 3\), both equations are satisfied.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental skill in algebra that allows you to visually represent equations on a coordinate plane. Each linear equation can be plotted as a straight line. The general form of a linear equation is usually written as \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. To accurately graph a line, you need at least two points. Here's how you can do it:

• Start by identifying the slope \(m\) and y-intercept \(b\) from the equation.
• Plot the y-intercept \((0, b)\) on the graph.
• Use the slope to find another point. For example, a slope of \(2\) means you go up 2 units and right 1 unit from the y-intercept.
• Draw a straight line through the points.

Graphing these equations makes it easier to see the solution visually as the intersection point of the lines.

Intersection Points of Graphs

The intersection point of two graphs is the point where the two lines meet on a coordinate plane. This point represents the solution to a system of equations because it satisfies both equations simultaneously. To find intersection points, you can graph each equation and identify where they cross.

• The point where the lines intersect gives you the values of \(x\) and \(y\) that solve both equations.
• If the lines intersect at one point, there is exactly one solution to the system.
• If the lines are parallel, they will not intersect, implying no solution.
• If the lines coincide (are the same line), there are infinitely many solutions.

In our exercise, after solving the equations, the intersection point comes out to be \((-2, 3)\). This means that \(x = -2\) and \(y = 3\) is the solution to the system.

Substitution Method in Algebra

The substitution method in algebra is a way to solve a system of equations by expressing one variable in terms of another and then substituting this expression into another equation. This method is particularly useful when one equation is already solved for one variable.Here's how you use the substitution method:

• Solve one of the equations for one of the variables. In the exercise, we solved for \(y\) in terms of \(x\): \(y = -1 - 2x\).
• Substitute this expression into the other equation. Replace \(y\) in the second equation with \(-1 - 2x\).
• Simplify and solve this equation to find the value of \(x\). In the exercise, this step gives \(x = -2\).
• Substitute the value of \(x\) back into the expression for \(y\) to find \(y\). In the exercise, this step yields \(y = 3\).
• Verify the solution by plugging \(x\) and \(y\) back into the original equations to check if they hold true.

Using substitution can simplify the process of finding the exact solution without graphing, especially when equations are complex.