Problem 25 - Precalculus: Mathematics for Calculus

Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6. $$\left\\{\begin{array}{r} x+3 y=5 \\ 2 x-y=3 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution is the ordered pair (2, 1).

Step 1: Write Down the Given System of Equations

The system of equations given is: \[ x + 3y = 5 \] \[ 2x - y = 3 \] Our goal is to find values of \(x\) and \(y\) that satisfy both equations simultaneously.

Step 2: Express One Variable in Terms of the Other

Let's solve the first equation for \(x\). From the equation \(x + 3y = 5\), solving for \(x\) gives:\[ x = 5 - 3y \] Now we have \(x\) expressed in terms of \(y\).

Step 3: Substitute the Expression into the Second Equation

Substitute \(x = 5 - 3y\) into the second equation:\[ 2(5 - 3y) - y = 3 \] This substitution will allow us to solve for \(y\).

Step 4: Simplify and Solve for y

Simplify the equation:\[ 10 - 6y - y = 3 \] Combine like terms:\[ 10 - 7y = 3 \] Subtract 10 from both sides:\[ -7y = 3 - 10 \] \[ -7y = -7 \] Divide by -7:\[ y = 1 \] Thus, \(y = 1\).

Step 5: Substitute y Back into x Expression

With \(y = 1\), substitute back into the expression we found for \(x\):\[ x = 5 - 3(1) \] \[ x = 5 - 3 \] \[ x = 2 \] Thus, \(x = 2\).

Step 6: Write the Solution as an Ordered Pair

The solution to the system is the ordered pair \((x, y) = (2, 1)\). This means the values of \(x\) and \(y\) that satisfy both equations are 2 and 1, respectively.

Step 7: Verify the Solution

Substitute \(x = 2\) and \(y = 1\) back into both original equations to verify:First equation:- \(2 + 3(1) = 5\), which is true.Second equation:- \(2(2) - 1 = 3\), which is also true.Thus, the solution \((2, 1)\) is verified.

Key Concepts

Linear Equations

Linear equations are a fundamental component of algebra where each term is either a constant or the product of a constant and a single variable. In our exercise, we have two linear equations as part of a system:

• \( x + 3y = 5 \)
• \( 2x - y = 3 \)

These equations are called 'linear' because they graph as straight lines in a coordinate plane. The goal when dealing with linear equations in a system is to find a set of values for the variables (in this case, \(x\) and \(y\)) that satisfy both equations at the same time. In simpler terms, we're looking for the point where the two lines intersect.

Substitution Method

The substitution method is a technique used for solving systems of equations by solving one equation for one variable and then substituting this expression into the other equation. In our example, we started with:1. Solving the first equation \( x + 3y = 5 \) for \(x\) gives us \( x = 5 - 3y \).2. This expression for \(x\) is then substituted into the second equation: \( 2x - y = 3 \).By substituting, we effectively reduce the problem from two equations in two unknowns to a single equation in one unknown, which is a simpler problem to solve. This method works well when one of the equations is easily solvable for one of the variables.

Ordered Pair Solution

Once we find the values of \(x\) and \(y\), we can express the solution as an ordered pair \((x, y)\). This ordered pair represents the values of \(x\) and \(y\) that when plugged into the original equations, make both true.In our exercise, after using the substitution method, we determined that \( y = 1 \) and \( x = 2 \). So the solution to this system of equations is \((2, 1)\).The concept of ordered pairs is crucial because it provides a clear and concise way to demonstrate the relationship between the variables that satisfy the given system of equations.

Verification of Solution

Verification is an important step to ensure that the solution found is correct. It involves substituting the values of \(x\) and \(y\) back into the original equations to ensure they hold true.For the solution \((2, 1)\) of our exercise, we substitute into both equations:

• First equation: \(2 + 3(1) = 5\). This simplifies to \(5 = 5\), which is correct.
• Second equation: \(2(2) - 1 = 3\). This simplifies to \(3 = 3\), which is also correct.

By confirming the calculations in both equations, we can confidently state that the solution is verified. Verifying is a critical step, ensuring the accuracy of the solution in all mathematical problems.