Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system by the method of substitution. Use a graphing utility to verify your results. $$\left\\{\begin{aligned} x+y &=0 \\ 4 x+3 y &=10 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 10\), \(y = -10\).

Step 1: Isolate one variable

Isolate \(x\) in the first equation. The first equation, \(x + y = 0\), can be rearranged to give \(x = -y\).

Step 2: Substitute and solve

Substitute the value of this expression \(x = -y\) into the second equation: \(4(-y) + 3y = 10\), which simplifies to \(y = -10\). Now substitue \(y = -10\) into our first equation to solve for \(x\): \(x + (-10) = 0\), which simplifies to \(x = 10\).

Step 3: Verification

Substitute \(x = 10\) and \(y = -10\) back into the original equations to verify. The checks out for both equations. Use a graphing utility to graph the two equations. The point of intersection ought to be (10,-10), which confirms our calculated solution.

Key Concepts

Understanding the Substitution Method

The substitution method is a useful technique for solving systems of linear equations, especially when one equation can be easily manipulated to express one variable in terms of the other. It involves a few straightforward steps:

• First, you isolate one of the variables in one of the equations. In our example, we started with the equation \(x + y = 0\), which easily rearranges to \(x = -y\).
• Next, substitute the expression obtained for the isolated variable into the other equation. In our system, substituting \(x = -y\) into the second equation \(4x + 3y = 10\) allows us to solve for \(y\).
• This substitution reduces the system from two equations in two variables to a single equation in one variable, making it much simpler to solve. For our system, it resulted in the equation \(4(-y) + 3y = 10\), ultimately leading to \(y = -10\).
• Finally, plug the value you found back into the expression for the isolated variable to find the second variable. Here, substituting \(y = -10\) into \(x = -y\) yields \(x = 10\).

This method is handy as it transforms a system of equations into a sequence of simpler equations.

Using Graphing Utilities for Verification

Graphing utilities are powerful tools that can be used to visualize systems of equations and verify solutions obtained algebraically. Let's see how this works:

• By entering the equations into a graphing utility, like a graphing calculator or software such as Desmos, you can graphically observe the behavior and intersection of the equations.
• For our solved system, we input the equations \(x + y = 0\) and \(4x + 3y = 10\). The intersection of these lines represents the solution to the system of equations.
• Once plotted, you can coalesce both the equations visually and the point where they intersect on the graph should match your calculated solution. In our example, this point of intersection is \((10, -10)\).

Utilizing a graphing utility not only confirms our answer but also provides better insight into how the equations interact in a visual space, hence reinforcing the algebraic computations.

Exploring Linear Equations

Linear equations are a fundamental concept in algebra, representing straight lines when graphed. They are expressed in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. Let's take a closer look:

• In our system, we had: \(x + y = 0\) and \(4x + 3y = 10\). Both are linear equations because the variables \(x\) and \(y\) are only to the first power, ensuring the graph of each equation is a straight line.
• The solution to a system of two linear equations in two unknowns, if it exists, is the point where their lines intersect.
• Different systems can have different types of solutions: one solution (the lines intersect at one point), no solution (the lines are parallel and never intersect), or infinitely many solutions (the lines are identical).

Understanding linear equations is crucial as they form the building blocks for more complex equations and systems in mathematics.