Chapter 7.1, Problem 8E

(a)

To determine

To check:Whether the ordered pair $\left(2,-13\right)$ is the solution of the system of equations $\{\begin{array}{c}4x^2+y=3\\ -x-y=11\end{array}$ .

Answer to Problem 8E

The ordered pair $\left(2,-13\right)$ is the solution of the system of the equations $\{\begin{array}{c}4x^2+y=3\\ -x-y=11\end{array}$ .

Explanation of Solution

Given information:

The given ordered pair is $\left(2,-13\right)$ and the system of equations is $\{\begin{array}{c}4x^2+y=3\\ -x-y=11\end{array}$ .

Calculation:

Let’s the ordered pair is $\left(2,-13\right)$ . To check the order pair is a solution of the system of equation, solve the given equation for $y$ by substituting the value of $x$ .

Consider the first equation $4x^2+y=3$ . Substitute $2$ for xand solve for y .

  $\begin{array}{c}4{\left(2\right)}^2+y=3\\ 16+y=3\\ y=3-16\\ =-13\end{array}$

From the above, it is clear that the value of $y$ is $-13$ which is not equal to $-13$ . Thus, the ordered pair $\left(2,-13\right)$ is a solution of the equation $4x^2+y=3$ .

Similarly, consider the second equation $-x-y=11$ . Substitute $2$ for xand solve for y .

  $\begin{array}{c}-2-y=11\\ y=-2-11\\ y=-13\end{array}$

From the above, it is clear that the value of $y$ is $-13$ which is not equal to $-13$ . Thus, the ordered pair $\left(2,-13\right)$ is a solution of the equation $-x-y=11$ .

The ordered pair can be a solution of the system of equations if it satisfied both the equations.

Therefore, the ordered pair $\left(2,-13\right)$ is the solution of the system of the equations $\{\begin{array}{c}4x^2+y=3\\ -x-y=11\end{array}$ .

(b)

To determine

To check:Whether the ordered pair $\left(-2,-9\right)$ is the solution of the system of equations $\{\begin{array}{c}4x^2+y=3\\ -x-y=11\end{array}$ .

Answer to Problem 8E

The ordered pair $\left(-2,-9\right)$ is not the solution of the system of the equations $\{\begin{array}{c}4x^2+y=3\\ -x-y=11\end{array}$ .

Explanation of Solution

Given information:

The given ordered pair is $\left(-2,-9\right)$ and the system of equations is $\{\begin{array}{c}4x^2+y=3\\ -x-y=11\end{array}$ .

Calculation:

Let’s the ordered pair is $\left(-2,-9\right)$ . To check the order pair is a solution of the system of equation, solve the given equation for $y$ by substituting the value of $x$ .

Consider the first equation $4x^2+y=3$ . Substitute $-2$ for xand solve for y .

  $\begin{array}{c}4{\left(-2\right)}^2+y=3\\ 16+y=3\\ y=3-16\\ =-13\end{array}$

From the above, it is clear that the value of $y$ is $-13$ which is not equal to $-9$ . Thus, the ordered pair $\left(-2,-9\right)$ is not a solution of the equation $4x^2+y=3$ .

Similarly, consider the second equation $-x-y=11$ . Substitute $-2$ for xand solve for y .

  $\begin{array}{c}2-y=11\\ y=2-11\\ =-9\end{array}$

From the above, it is clear that the value of $y$ is $-9$ which is equal to $-9$ . Thus, the ordered pair $\left(-2,-9\right)$ is a solution of the equation $-x-y=11$ .

The ordered pair can be a solution of the system of equations if it satisfied both the equations.

Therefore, the ordered pair $\left(-2,-9\right)$ is not the solution of the system of the equations $\{\begin{array}{c}4x^2+y=3\\ -x-y=11\end{array}$ .

(c)

To determine

To check:Whether the ordered pair $\left(-\frac{3}{2},6\right)$ is the solution of the system of equations $\{\begin{array}{c}4x^2+y=3\\ -x-y=11\end{array}$ .

Answer to Problem 8E

The ordered pair $\left(-\frac{3}{2},6\right)$ is not the solution of the system of the equations $\{\begin{array}{c}4x^2+y=3\\ -x-y=11\end{array}$ .

Explanation of Solution

Given information:

The given ordered pair is $\left(-\frac{3}{2},6\right)$ and the system of equations is $\{\begin{array}{c}4x^2+y=3\\ -x-y=11\end{array}$ .

Calculation:

Let’s the ordered pair is $\left(-\frac{3}{2},6\right)$ To check the order pair is a solution of the system of equation, solve the given equation for $y$ by substituting the value of $x$ .

Consider the first equation $4x^2+y=3$ . Substitute $-\frac{3}{2}$ for xand solve for y .

  $\begin{array}{c}4{\left(-\frac{3}{2}\right)}^2+y=3\\ 9+y=3\\ y=3-9\\ =-6\end{array}$

From the above, it is clear that the value of $y$ is $-6$ which is not equal to $6$ . Thus, the ordered pair $\left(-\frac{3}{2},6\right)$ is not a solution of the equation $4x^2+y=3$ .

Similarly, consider the second equation $-x-y=11$ . Substitute $-\frac{3}{2}$ for xand solve for y .

  $\begin{array}{c}\frac{3}{2}-y=11\\ y=\frac{3}{2}-11\\ y=\frac{3-22}{2}\\ =-\frac{19}{3}\end{array}$

From the above, it is clear that the value of $y$ is $-\frac{19}{3}$ which is not equal to $6$ . Thus, the ordered pair $\left(-\frac{3}{2},6\right)$ is not a solution of the equation $-x-y=11$ .

The ordered pair can be a solution of the system of equations if it satisfied both the equations.

Therefore, the ordered pair $\left(-\frac{3}{2},6\right)$ is not the solution of the system of the equations $\{\begin{array}{c}4x^2+y=3\\ -x-y=11\end{array}$ .

(d)

To determine

To check:Whether the ordered pair $\left(-\frac{7}{4},-\frac{37}{4}\right)$ is the solution of the system of equations $\{\begin{array}{c}4x^2+y=3\\ -x-y=11\end{array}$ .

Answer to Problem 8E

The ordered pair $\left(-\frac{7}{4},-\frac{37}{4}\right)$ is the solution of the system of the equations $\{\begin{array}{c}4x^2+y=3\\ -x-y=11\end{array}$ .

Explanation of Solution

Given information:

The given ordered pair is $\left(-\frac{7}{4},-\frac{37}{4}\right)$ and the system of equations is $\{\begin{array}{c}4x^2+y=3\\ -x-y=11\end{array}$ .

Calculation:

Let’s the ordered pair is $\left(-\frac{7}{4},-\frac{37}{4}\right)$ To check the order pair is a solution of the system of equation, solve the given equation for $y$ by substituting the value of $x$ .

Consider the first equation $4x^2+y=3$ . Substitute $-\frac{7}{4}$ for xand solve for y .

  $\begin{array}{c}4{\left(-\frac{7}{4}\right)}^2+y=3\\ \frac{49}{4}+y=3\\ y=3-\frac{49}{4}\\ =-\frac{37}{4}\end{array}$

From the above, it is clear that the value of $y$ is $-\frac{37}{4}$ which is equal to $-\frac{37}{4}$ . Thus, the ordered pair $\left(-\frac{7}{4},-\frac{37}{4}\right)$ is a solution of the equation $4x^2+y=3$ .

Similarly, consider the second equation $-x-y=11$ . Substitute $-\frac{7}{4}$ for xand solve for y .

  $\begin{array}{c}\frac{7}{4}-y=11\\ y=\frac{7}{4}-11\\ y=\frac{7-44}{4}\\ =-\frac{37}{4}\end{array}$

From the above, it is clear that the value of $y$ is $-\frac{37}{4}$ which is equal to $-\frac{37}{4}$ . Thus, the ordered pair $\left(-\frac{7}{4},-\frac{37}{4}\right)$ is a solution of the equation $-x-y=11$ .

The ordered pair can be a solution of the system of equations if it satisfied both the equations.

Therefore, the ordered pair $\left(-\frac{7}{4},-\frac{37}{4}\right)$ is the solution of the system of the equations $\{\begin{array}{c}4x^2+y=3\\ -x-y=11\end{array}$ .