Problem 2 - Algebra and Trigonometry

Question

Use the substitution method to find all solutions of the system of equations. \(\left\\{\begin{array}{c}{2 x+y=7} \\ {x+2 y=2}\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system is \((x, y) = (4, -1)\).

Step 1: Solve one equation for one variable

Start by solving the first equation \(2x + y = 7\) for \(y\). \[ y = 7 - 2x \]

Step 2: Substitute for the other variable in the second equation

Substitute the expression for \(y\) from Step 1 into the second equation \(x + 2y = 2\): \[ x + 2(7 - 2x) = 2 \] This simplifies to: \[ x + 14 - 4x = 2 \]

Step 3: Simplify and solve for one variable

Rearrange the equation from Step 2 to find \(x\): \[ -3x + 14 = 2 \] Subtract 14 from both sides: \[ -3x = -12 \] Divide each side by -3: \[ x = 4 \]

Step 4: Substitute back to find the other variable

Now substitute \(x = 4\) back into the equation for \(y\) from Step 1: \[ y = 7 - 2(4) \] This gives: \[ y = 7 - 8 = -1 \]

Step 5: Verify the solution

Verify the solution \((x, y) = (4, -1)\) by substituting it back into both original equations:1. \(2(4) + (-1) = 8 - 1 = 7\)2. \(4 + 2(-1) = 4 - 2 = 2\) Both equations are satisfied, confirming our solution is correct.

Key Concepts

Solving Systems of Equations

When we talk about solving systems of equations, we're diving into how multiple equations can work together to find a common solution. A system of equations usually involves two or more equations that share common variables. The goal is to find the values of these variables that satisfy all the equations in the system.
In the problem above, we were given a system of two linear equations. The solution consists of an ordered pair \(x, y\) that will balance both equations at the same time.
There are various methods to solve these systems, such as graphing, substitution, and elimination. But, in this case, we'll focus on the substitution method which is a highly effective algebraic approach.

Algebraic Substitution

Algebraic substitution is a method used in solving systems of equations by replacing one variable with an expression obtained from another equation. This involves a few steps:

• First, solve one of the equations for one variable.
• Next, substitute this expression into the other equation.

With our problem, we started with the equation \(2x + y = 7\). We solved for \y\, getting \(y = 7 - 2x\). Then, we replaced \y\ in the second equation \(x + 2y = 2\) with \(7 - 2x\). By substituting, we replaced one equation with a single-variable equation, which makes it easier to solve.

Linear Equations

Linear equations are those equations where each term is either a constant or the product of a constant and a single variable. They are called "linear" because, when graphed, they make a straight line.
In our exercise, the given system is composed of two linear equations: \(2x + y = 7\) and \(x + 2y = 2\).
These equations collectively formed a system of equations that we aimed to solve using substitution. As you solve, the linearity allows us to go step-by-step with simple algebraic manipulations such as substitution and simplification.

Solution Verification in Algebra

Once you've computed a solution, it’s crucial to verify that it's correct. This involves plugging the values back into the original equations to see if they hold true.
In our case, after finding \(x = 4\) and \(y = -1\), we substituted these values back into both of the starting equations: \(2x + y = 7\) and \(x + 2y = 2\).
Both substitutions confirmed that left sides of the equations equal the right sides, verifying the solution.
Verification ensures accuracy and prevents potential errors during calculations.