Problem 8 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether each ordered pair is a solution of the system of equations. $$\left\\{\begin{array}{l} 4 x^{2}+y=3 \\ -x-y=11 \end{array}\right.$$ (a) (2,-13) (b) (-2,-9) (c) \(\left(-\frac{3}{2}, 6\right)\) (d) \(\left(-\frac{7}{4},-\frac{37}{4}\right)\)

Step-by-Step Solution

Answer

The ordered pairs (2,-13) and \(-\frac{7}{4},-\frac{37}{4}\) are solutions to the system of equations. The pairs (-2,-9) and \(-\frac{3}{2}, 6\) are not solutions.

Step 1: Check the ordered pair (2,-13)

Substitute \(x = 2\) and \(y = -13\) into both equations. The first equation becomes \(4*(2)^2 + (-13) = 3\), which simplifies to \(16 - 13 = 3\). This holds true. The second equation becomes \(-2 - (-13) = 11\), which simplifies to \(11 = 11\). This also holds true. Thus, the pair (2,-13) is a solution of the system.

Step 2: Check the ordered pair (-2,-9)

Substitute \(x = -2\) and \(y = -9\) into both equations. The first equation becomes \(4*(-2)^2 + (-9) = 3\), which simplifies to \(16 - 9 \neq 3\). This does not hold true. So, the pair (-2,-9) is not a solution of the system.

Step 3: Check the ordered pair \(-\frac{3}{2}, 6\)

Substitute \(x = -\frac{3}{2}\) and \(y = 6\) into both equations. The first equation becomes \(4*(-\frac{3}{2})^2 + 6 = 3\), which simplifies to \(9 \neq 3\). This does not hold true. So, the pair \(-\frac{3}{2}, 6\) is not a solution of the system.

Step 4: Check the ordered pair \(-\frac{7}{4},-\frac{37}{4}\)

Substitute \(x = -\frac{7}{4}\) and \(y = -\frac{37}{4}\) into both equations. The first equation becomes \(4*(-\frac{7}{4})^2 + (-\frac{37}{4}) = 3\), which simplifies to \(49/4 - 37/4 = 3\). This holds true. The second equation becomes \(7/4 + \frac{37}{4} = 11\), which simplifies to \(11 = 11\). This also holds true. Thus, the pair \(-\frac{7}{4},-\frac{37}{4}\) is a solution of the system.

Key Concepts

Ordered Pairs

In mathematics, ordered pairs are crucial for identifying positions on the Cartesian coordinate system. Each ordered pair is written in the form

• \((x, y)\)

where \(x\) represents the horizontal component, and \(y\) the vertical component.
These pairs are used to determine solutions for systems of equations, which may involve two or more equations with the same variables.
Checking if an ordered pair is a solution involves substituting the pair into each equation.
For instance, if given the equations:

• \(4x^2 + y = 3\)
• \(-x - y = 11\)

we substitute the \(x\) and \(y\) from the pair into the equations to check for true statements in both cases.
When evaluated for both equations and they hold true, the pair is a solution to the system.

Solution Verification

Solution verification involves substituting the values of the ordered pairs into the system of equations to see if both equations are satisfied.
Here's how it works: substitute the \(x\) value in place of \(x\) and the \(y\) value in place of \(y\) in each equation.

• If both equations return true statements (i.e., left-hand side equals the right-hand side), the ordered pair is a solution.
• If even one equation does not hold true, then the pair is not a solution.

This verification step is essential because it ensures that the selected solution indeed satisfies the entire system, providing accuracy in problem-solving.
In the given exercise, pairs like

• (2, -13) and
• \((-\frac{7}{4}, -\frac{37}{4})\)

held true for both equations, verifying them as solutions.

Substitution Method

The substitution method is a technique primarily used to solve systems of equations by finding the solutions through substitution. This method involves several steps:

• Choose one equation and solve it for one variable in terms of the other, such as \(x\) in terms of \(y\) or vice versa.
• Substitute this expression into the other equation.
• Solve the resulting equation for the other variable.
• Plug this solution back into the expression to find the value of the first variable.

By substituting values, we find exact points where the equations intersect.
For instance, in the first equation \(4x^2 + y = 3\), if we solve for \(y\) and substitute in the second equation \(-x - y = 11\), this can help verify whether an ordered pair is correct.
This method systematically reduces the complexity of solving by focusing on one variable at a time.