Problem 20 - Intermediate Algebra

Question

Determine whether the ordered pair is a solution of the system of equations. See Example 1. $$ (2.1,-3.2) ;\left\\{\begin{array}{c} x+y=-1.1 \\ 2 x-3 y=13.8 \end{array}\right. $$

Step-by-Step Solution

Answer

The ordered pair \((2.1, -3.2)\) is a solution to the system of equations.

Step 1: Understand the Problem

We are given a system of two equations with two variables, \(x\) and \(y\). The ordered pair \((2.1, -3.2)\) needs to be checked for both equations to determine if it satisfies the system.

Step 2: Substitute into First Equation

The first equation is \(x + y = -1.1\). Substitute \(x = 2.1\) and \(y = -3.2\) into this equation: \[ 2.1 + (-3.2) \stackrel{?}{=} -1.1 \] Calculate the left side: \[ 2.1 - 3.2 = -1.1 \] Both sides are equal, so the ordered pair satisfies the first equation.

Step 3: Substitute into Second Equation

The second equation is \(2x - 3y = 13.8\). Again, substitute \(x = 2.1\) and \(y = -3.2\): \[ 2(2.1) - 3(-3.2) \stackrel{?}{=} 13.8 \] Calculate each term: \[ 4.2 + 9.6 = 13.8 \] Since both sides are equal, the ordered pair also satisfies the second equation.

Step 4: Conclusion

Since the ordered pair \((2.1, -3.2)\) satisfies both equations in the system, it is a solution to the system of equations.

Key Concepts

Ordered Pairs: A Basic Building Block

Ordered pairs are fundamental in understanding how to locate points on a plane and evaluate solutions in systems of equations. An ordered pair typically takes the form \((x, y)\) with the first number, \(x\), representing the horizontal position, and the second number, \(y\), representing the vertical position.

• In the context of a coordinate system, these numbers indicate a specific point on a graph.
• This concept is crucial for understanding solutions to equations since they can be visually represented as intersections of lines or curves on a graph.
• Each equation in a system introduces constraints that determine acceptable values for \(x\) and \(y\). Together, these constraints define the set of possible solutions.

Mastering ordered pairs will enhance your ability to solve systems of equations and graph mathematical concepts effectively.

Substitution Method: Solving Systems Step-by-Step

The substitution method is a popular algebraic technique used to find the solution to a system of equations. It involves replacing one variable with an equivalent expression obtained from another equation, essentially transforming a two-variable problem into a one-variable problem. This method simplifies the process of finding values for \(x\) and \(y\) that satisfy both equations.

• First, solve one of the equations for one variable. For instance, solve for \(y\) in terms of \(x\) using the first equation.
• Next, substitute this expression into the other equation. This step involves plugging the expression derived for \(y\) into the second equation.
• Solve the resulting single-variable equation. This will give a specific value for \(x\).
• Finally, substitute back to find the value of the other variable. Use the calculated \(x\) value to find \(y\).

Once you have both values, you can confirm these ordered pairs as solutions by plugging them back into the original equations.

Identifying Solutions to Equations

A solution to a system of equations is a specific set of values for variables that makes all equations in the system true simultaneously. In simple terms, it means finding where the graphs of these equations intersect if plotted.

• Once you have determined potential solutions using methods like substitution, verify by substituting these values back into each original equation.
• If both equations are satisfied, the ordered pair is indeed a solution to the system.
• Remember, a system can have one solution, no solution, or infinitely many solutions, depending on how equations relate to one another.

Checking the validity of your solutions is a crucial final step in solving systems of equations. This ensures that the solutions logically fit all mathematical requirements indicated by the equations.