Problem 1 - Algebra and Trigonometry

Question

The system of equations $$\left\\{\begin{array}{l}{2 x+3 y=7} \\ {5 x-y=9}\end{array}\right.$$ is a system of two equations in the two variables _____ and _____ To determine whether \((5,-1)\) is a solution of this system, we check whether \(x=5\) and \(y=-1\) satisfy each _____ in the system. Which of the following are solutions of this system? $$(5,-1), \quad(-1,3), \quad(2,1)$$

Step-by-Step Solution

Answer

The solution to the system is (2, 1).

Step 1: Understand the Problem

We have a system of two equations in two variables, \(x\) and \(y\). The given system is \(\{2x + 3y = 7,\> 5x - y = 9\}\). We need to determine if any of the given pairs \((5, -1)\), \((-1, 3)\), and \((2, 1)\) are solutions.

Step 2: Substitute \((5, -1)\) into the System

To check if \((5, -1)\) is a solution, substitute \(x = 5\) and \(y = -1\) into both equations. 1. For the first equation, \(2(5) + 3(-1) = 10 - 3 = 7\), which is true.2. For the second equation, \(5(5) - (-1) = 25 + 1 = 26\), which is not equal to 9.Therefore, \((5, -1)\) is not a solution.

Step 3: Substitute \((-1, 3)\) into the System

Substitute \(x = -1\) and \(y = 3\) into both equations.1. For the first equation, \(2(-1) + 3(3) = -2 + 9 = 7\), which is true.2. For the second equation, \(5(-1) - 3 = -5 - 3 = -8\), which is not equal to 9.Therefore, \((-1, 3)\) is not a solution.

Step 4: Substitute \((2, 1)\) into the System

Substitute \(x = 2\) and \(y = 1\) into both equations.1. For the first equation, \(2(2) + 3(1) = 4 + 3 = 7\), which is true.2. For the second equation, \(5(2) - 1 = 10 - 1 = 9\), which is true.Therefore, \((2, 1)\) is a solution.

Key Concepts

Substitution Method

The substitution method is a key technique for solving systems of equations. It involves solving one of the equations for one variable in terms of the other variable. Then, this expression is substituted into the other equation. This method reduces the system from two equations to one equation with a single variable. Here is how it works step by step:

• Select one of the equations and solve for one of the variables. For instance, if you have the equation \(2x + 3y = 7\), you might solve for \(x\) in terms of \(y\): \(x = \frac{7 - 3y}{2}\).
• Substitute the expression from the first equation into the second equation. For the other equation \(5x - y = 9\), you would replace \(x\) with \(\frac{7 - 3y}{2}\) to have one equation with only \(y\).
• Solve the new single-variable equation to find the value of the chosen variable. Once you know one variable, use it to find the value of the other variable using the expression from the substitution.
• Verify this solution by substituting back into the original equations to check if both are satisfied.

This method is particularly useful when one of the variables is easy to isolate. It simplifies the process and allows us to find the solution in a more straightforward manner.

Solution Verification

Verifying a solution for a system of equations is a critical step, ensuring that the proposed solution actually satisfies both equations. Let's break down the process:

• Once you find potential solutions like \((5, -1), (-1, 3), (2, 1)\), substitute them into each equation of the system to confirm each equation holds true.
• For example, if we suppose \((x, y) = (2, 1)\), plug \(x = 2\) and \(y = 1\) into each equation, \(2(2) + 3(1) = 7\) and \(5(2) - 1 = 9\). If both expressions simplify to true statements, then \((2, 1)\) is a valid solution.
• If a pair makes both equations true, it is the correct solution. If not, like \((5, -1)\) giving \(26\) in the second equation instead of \(9\), it is not a solution.

Verifying solutions is essential because it confirms the calculations are correct and that the solutions meet all requirements imposed by the system of equations.

Linear Equations

Linear equations are mathematical statements of equality involving constants and variables with no exponents higher than one. Unlike quadratic or other polynomial equations, linear equations graph as straight lines. Consider the basic form of a linear equation: \(ax + by = c\). Here:

• \(x\) and \(y\) are variables.
• \(a\) and \(b\) are coefficients.
• \(c\) is a constant.

In a system of linear equations, we work with two or more linear equations containing the same set of variables. A solution is a set of values for the variables that satisfy all equations simultaneously.For instance, our given system \(\{2x + 3y = 7, 5x - y = 9\}\) is a typical example. Solving these equations involves finding the \( (x, y) \) that makes both true, such as \((2, 1)\) as verified.Linear equations are fundamental because they model numerous real-world situations and allow us to predict outcomes and relationships efficiently through their straightforward structure.