Problem 8 - Algebra and Trigonometry

Question

\(5-8\) Use the substitution method to find all solutions of the system of equations. $$ \left\\{\begin{array}{l}{2 x+y=7} \\ {x+2 y=2}\end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = 4\) and \(y = -1\).

Step 1: Choose an Equation to Solve for a Variable

We begin by isolating one variable in one of the equations. Let's take the first equation, \(2x + y = 7\), and solve for \(y\). To do this, we subtract \(2x\) from both sides: \(y = 7 - 2x\).

Step 2: Substitute the Expression into the Other Equation

Now that we have \(y\) in terms of \(x\), substitute \(y = 7 - 2x\) into the second equation \(x + 2y = 2\). This gives us: \(x + 2(7 - 2x) = 2\).

Step 3: Simplify and Solve the Equation for x

Expand and combine like terms in the equation from Step 2: \(x + 14 - 4x = 2\). Simplify this to \(-3x + 14 = 2\). Then, subtract 14 from both sides to get \(-3x = -12\). Divide both sides by \(-3\) to find \(x = 4\).

Step 4: Substitute x Back to Find y

Use the value of \(x = 4\) in the equation \(y = 7 - 2x\). Substitute \(x\) with 4: \(y = 7 - 2(4) = 7 - 8 = -1\). So, \(y = -1\).

Step 5: Verify the Solution

Substitute \(x = 4\) and \(y = -1\) back into the original equations to verify the solution. For \(2x + y = 7\): \(2(4) - 1 = 8 - 1 = 7\), and for \(x + 2y = 2\): \(4 + 2(-1) = 4 - 2 = 2\). Both equations are satisfied, confirming the solution is correct.

Key Concepts

Understanding System of Equations

A system of equations is a collection of two or more equations with the same set of variables. The purpose of solving a system is to find the values of the variables that satisfy all equations at the same time. In the given problem, the system consists of two linear equations involving the variables \(x\) and \(y\):

• \(2x + y = 7\)
• \(x + 2y = 2\)

These equations graph as straight lines on a coordinate plane, and the solution to the system is the point where these two lines intersect. Finding this intersection point means we are finding the values of \(x\) and \(y\) that make both equations true simultaneously. This concept underlies many fields of problem-solving in mathematics, physics, engineering, and economics.
The key thing to remember is that a system of equations either has a single solution (intersection point), no solution (parallel lines), or infinitely many solutions (coincident lines). In this exercise, we aim to find the unique solution where both lines intersect.

Steps to Solve Linear Equations

Solving linear equations involves finding the values of variables that make the equality true. For a single equation, this can often be done by simple algebraic manipulations:

• Combining like terms
• Using addition or subtraction to isolate the variable
• Using multiplication or division to simplify the equation

Within a system of equations, each equation adds another layer of complexity, but the fundamental approach remains the same. In our system, we first focused on the equation \(2x + y = 7\) to isolate one variable, \(y\). This process involved rearranging the equation to make \(y\) the subject by subtracting \(2x\) from both sides, resulting in \(y = 7 - 2x\).
This simplification allows us to proceed to the next step within a system context, setting up for substitution in the other equation, while dealing with one variable at a time. Solving linear equations is the core skill needed to tackle more complex mathematical problems effectively.

Exploring Algebraic Substitution

Algebraic substitution is a method used to solve systems of equations by replacing one variable with an expression containing the other variable. This method is particularly useful when a variable is already isolated, or can be isolated easily in one of the equations. In our exercise, after isolating \(y\) in the first equation, we substitute \(y = 7 - 2x\) into the second equation \(x + 2y = 2\).
This substitution step transforms the problem into a single equation in one variable: \(x + 2(7 - 2x) = 2\). The equation can then be simplified and solved for \(x\). Once \(x\) is known, substituting it back into the expression for \(y\) provides the solution for the other variable.

• Simplifies a system of equations into a single equation
• Easier to manage once manipulated
• Can be applied to more complicated systems

This method is highly effective as it reduces the complexity of the original system of equations, making it much easier to find the solution. Solving using substitution not only aids in comprehension but also in reinforcing foundational algebra skills.