Question

Determine whether each ordered pair is a solution of the system of equations. \(\left\\{\begin{array}{rr}x+4 y= & -3 \\ 5 x-y= & 6\end{array}\right.\) (a) \((-1,-1)\) (b) \((1,-1)\)

Short answer

The ordered pair \((-1, -1)\) is not a solution to the system of equations, whereas the ordered pair \((1, -1)\) is a solution.

Step-by-step solution

Substitute Ordered Pair (a) into Both Equations

Take the ordered pair (a) \((-1, -1)\) and substitute \(x=-1\) and \(y=-1\) into each equation. For the first equation, it will look like \((-1) + 4*(-1) = -3\), and for the second equation, it will look like \(5*(-1) - (-1) = 6\).

Verify If Both Equations Hold True for Ordered Pair (a)

Now, check these results. The first equation simplifies to \(-1 - 4 = -5\), which is not equal to \(-3\). The second equation simplifies to \(-5 + 1 = -4\), which is not equal to \(6\). Therefore, the ordered pair \((-1, -1)\) is not a solution to the system of equations.

Substitute Ordered Pair (b) into Both Equations

Next, take the ordered pair (b) \((1, -1)\) and substitute \(x=1\) and \(y=-1\) into each equation. For the first equation, it will look like \(1 + 4*(-1) = -3\), and for the second equation, it will look like \(5*1 - (-1) = 6\).

Verify If Both Equations Hold True for Ordered Pair (b)

Now, check these results. The first equation simplifies to \(1 - 4 = -3\), which is equal to \(-3\). The second equation simplifies to \(5 + 1 = 6\), which is equal to \(6\). Therefore, the ordered pair \((1, -1)\) is a solution to the system of equations.

Key concepts

Ordered Pairs

An ordered pair is a fundamental concept in mathematics, especially when dealing with systems of equations. In simplest terms, an ordered pair consists of two elements that are written in a specific sequence inside parentheses, such as \((x, y)\). The first element is known as the x-coordinate, and the second as the y-coordinate. These pairs are essential for identifying positions or points on a coordinate plane.

When solving systems of equations, ordered pairs are tested to determine if they satisfy all equations in the system. If they do, that particular pair is a solution to the system. In our example, we evaluated two ordered pairs:

• \((-1, -1)\)
• \((1, -1)\)

By substituting these pairs into the given equations, we checked whether they satisfy the system or not.

Solution Verification

Solution verification is crucial when working with systems of equations. It involves substituting an ordered pair into each equation of the system to determine if the pair satisfies every equation. Only pairs that make all equations true are valid solutions.

In the exercise, we verified ordered pair \((-1, -1)\) and found it's not a solution because:

• First equation: \(-1 + 4(-1) = -1 - 4 = -5\), which isn't equal to \(-3\).
• Second equation: \(5(-1) - (-1) = -5 + 1 = -4\), which isn't equal to \(6\).

Thus, it didn’t satisfy all equations. Conversely, for \((1, -1)\), both equations were satisfied:

• First equation: \(1 + 4(-1) = 1 - 4 = -3\), which equals \(-3\).
• Second equation: \(5(1) - (-1) = 5 + 1 = 6\), which equals \(6\).

Therefore, the ordered pair \((1, -1)\) is indeed a solution to the system of equations. Regularly double-checking solutions helps avoid errors and confirms accuracy.

Substitution Method

The substitution method is a popular technique for solving systems of equations. It comprises a few simple steps:

• First, solve one of the equations for one of the variables, expressing it in terms of the other variable.
• Second, substitute this expression into the other equation.
• Solve the resulting equation to find the value of one variable.
• Finally, substitute back to find the remaining variable's value.

This method is especially useful when the system has at least one simple equation, where isolating a variable is straightforward.

For our exercise, however, we used substitution differently by directly substituting values from ordered pairs into the equations to verify if they hold true. Either way, the substitution method is a handy tool in algebra, aiding in systematically finding solutions or verifying given pairs in a system.