Solved on Feb 28, 2024

Determine if each ordered pair $(x, y)$ is a solution to the system of linear equations $\begin{cases} 4x - 5y = 8\\ -3x + 2y = 1 \end{cases}$ .

STEP 1

Assumptions
1. We have a system of linear equations: $\left\{\begin{array}{c} 4 x-5 y=8 \\ -3 x+2 y=1 \end{array}\right.$
2. We need to check if each given ordered pair $(x, y)$ is a solution to the system.
3. To verify if an ordered pair is a solution, we will substitute $x$ and $y$ into both equations and check if the left-hand side equals the right-hand side.

STEP 2

Let's start with the first ordered pair $(-6, 2)$ . We will substitute $x = -6$ and $y = 2$ into the first equation $4x - 5y = 8$ .
$4(-6) - 5(2) = 8$

STEP 3

Calculate the left-hand side of the first equation with the substituted values.
$-24 - 10 = 8$

STEP 4

Check if the left-hand side equals the right-hand side.
$-34 \neq 8$
Since the left-hand side does not equal the right-hand side, the ordered pair $(-6, 2)$ is not a solution to the first equation.

STEP 5

We will now substitute $x = -6$ and $y = 2$ into the second equation $-3x + 2y = 1$ .
$-3(-6) + 2(2) = 1$

STEP 6

Calculate the left-hand side of the second equation with the substituted values.
$18 + 4 = 1$

STEP 7

Check if the left-hand side equals the right-hand side.
$22 \neq 1$
Since the left-hand side does not equal the right-hand side, the ordered pair $(-6, 2)$ is not a solution to the second equation either.

STEP 8

We can conclude that the ordered pair $(-6, 2)$ is not a solution to the system of equations. We mark "No" in the table for this ordered pair.

STEP 9

Next, let's check the ordered pair $(-3, -4)$ . We substitute $x = -3$ and $y = -4$ into the first equation $4x - 5y = 8$ .
$4(-3) - 5(-4) = 8$

STEP 10

Calculate the left-hand side of the first equation with the substituted values.
$-12 + 20 = 8$

STEP 11

Check if the left-hand side equals the right-hand side.
$8 = 8$
The left-hand side equals the right-hand side, so the ordered pair $(-3, -4)$ satisfies the first equation.

STEP 12

Now, substitute $x = -3$ and $y = -4$ into the second equation $-3x + 2y = 1$ .
$-3(-3) + 2(-4) = 1$

STEP 13

Calculate the left-hand side of the second equation with the substituted values.
$9 - 8 = 1$

STEP 14

Check if the left-hand side equals the right-hand side.
$1 = 1$
The left-hand side equals the right-hand side, so the ordered pair $(-3, -4)$ also satisfies the second equation.

STEP 15

We can conclude that the ordered pair $(-3, -4)$ is a solution to the system of equations. We mark "Yes" in the table for this ordered pair.

STEP 16

Now, let's check the ordered pair $(2, 0)$ . We substitute $x = 2$ and $y = 0$ into the first equation $4x - 5y = 8$ .
$4(2) - 5(0) = 8$

STEP 17

Calculate the left-hand side of the first equation with the substituted values.
$8 - 0 = 8$

STEP 18

Check if the left-hand side equals the right-hand side.
$8 = 8$
The left-hand side equals the right-hand side, so the ordered pair $(2, 0)$ satisfies the first equation.

STEP 19

Now, substitute $x = 2$ and $y = 0$ into the second equation $-3x + 2y = 1$ .
$-3(2) + 2(0) = 1$

STEP 20

Calculate the left-hand side of the second equation with the substituted values.
$-6 + 0 = 1$

STEP 21

Check if the left-hand side equals the right-hand side.
$-6 \neq 1$
Since the left-hand side does not equal the right-hand side, the ordered pair $(2, 0)$ is not a solution to the second equation.

STEP 22

We can conclude that the ordered pair $(2, 0)$ is not a solution to the system of equations. We mark "No" in the table for this ordered pair.

STEP 23

Finally, let's check the ordered pair $(3, 5)$ . We substitute $x = 3$ and $y = 5$ into the first equation $4x - 5y = 8$ .
$4(3) - 5(5) = 8$

STEP 24

Calculate the left-hand side of the first equation with the substituted values.
$12 - 25 = 8$

STEP 25

Check if the left-hand side equals the right-hand side.
$-13 \neq 8$
Since the left-hand side does not equal the right-hand side, the ordered pair $(3, 5)$ is not a solution to the first equation.

STEP 26

We will now substitute $x = 3$ and $y = 5$ into the second equation $-3x + 2y = 1$ .
$-3(3) + 2(5) = 1$

STEP 27

Calculate the left-hand side of the second equation with the substituted values.
$-9 + 10 = 1$

STEP 28

Check if the left-hand side equals the right-hand side.
$1 = 1$
The left-hand side equals the right-hand side, but since the ordered pair $(3, 5)$ did not satisfy the first equation, it is not a solution to the system.

STEP 29

We can conclude that the ordered pair $(3, 5)$ is not a solution to the system of equations. We mark "No" in the table for this ordered pair.
The completed table is as follows: \begin{tabular}{|c|c|c|} \hline \multirow{2}{*}{ $(x, y)$ } & \multicolumn{2}{|c|}{ Is it a solution? } \\ \cline { 2 - 3 } & Yes & No \\ \hline $(-6,2)$ & & X \\ \hline $(-3,-4)$ & X & \\ \hline $(2,0)$ & & X \\ \hline $(3,5)$ & & X \\ \hline \end{tabular}

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Determine if each ordered pair (x,y)(x, y)(x,y) is a solution to the system of linear equations {4x−5y=8−3x+2y=1\begin{cases} 4x - 5y = 8\\ -3x + 2y = 1 \end{cases}{4x−5y=8−3x+2y=1​.

STEP 1

STEP 2

Let's start with the first ordered pair (−6,2)(-6, 2)(−6,2). We will substitute x=−6x = -6x=−6 and y=2y = 2y=2 into the first equation 4x−5y=84x - 5y = 84x−5y=8.4(−6)−5(2)=84(-6) - 5(2) = 84(−6)−5(2)=8

STEP 3

Calculate the left-hand side of the first equation with the substituted values.−24−10=8-24 - 10 = 8−24−10=8

STEP 4

Check if the left-hand side equals the right-hand side.−34≠8-34 \neq 8−34=8Since the left-hand side does not equal the right-hand side, the ordered pair (−6,2)(-6, 2)(−6,2) is not a solution to the first equation.

STEP 5

We will now substitute x=−6x = -6x=−6 and y=2y = 2y=2 into the second equation −3x+2y=1-3x + 2y = 1−3x+2y=1.−3(−6)+2(2)=1-3(-6) + 2(2) = 1−3(−6)+2(2)=1

STEP 6

Calculate the left-hand side of the second equation with the substituted values.18+4=118 + 4 = 118+4=1

STEP 7

Check if the left-hand side equals the right-hand side.22≠122 \neq 122=1Since the left-hand side does not equal the right-hand side, the ordered pair (−6,2)(-6, 2)(−6,2) is not a solution to the second equation either.

STEP 8

We can conclude that the ordered pair (−6,2)(-6, 2)(−6,2) is not a solution to the system of equations. We mark "No" in the table for this ordered pair.

STEP 9

Next, let's check the ordered pair (−3,−4)(-3, -4)(−3,−4). We substitute x=−3x = -3x=−3 and y=−4y = -4y=−4 into the first equation 4x−5y=84x - 5y = 84x−5y=8.4(−3)−5(−4)=84(-3) - 5(-4) = 84(−3)−5(−4)=8

STEP 10

Calculate the left-hand side of the first equation with the substituted values.−12+20=8-12 + 20 = 8−12+20=8

STEP 11

Check if the left-hand side equals the right-hand side.8=88 = 88=8The left-hand side equals the right-hand side, so the ordered pair (−3,−4)(-3, -4)(−3,−4) satisfies the first equation.

STEP 12

Now, substitute x=−3x = -3x=−3 and y=−4y = -4y=−4 into the second equation −3x+2y=1-3x + 2y = 1−3x+2y=1.−3(−3)+2(−4)=1-3(-3) + 2(-4) = 1−3(−3)+2(−4)=1

STEP 13

Calculate the left-hand side of the second equation with the substituted values.9−8=19 - 8 = 19−8=1

STEP 14

Check if the left-hand side equals the right-hand side.1=11 = 11=1The left-hand side equals the right-hand side, so the ordered pair (−3,−4)(-3, -4)(−3,−4) also satisfies the second equation.

STEP 15

We can conclude that the ordered pair (−3,−4)(-3, -4)(−3,−4) is a solution to the system of equations. We mark "Yes" in the table for this ordered pair.

STEP 16

Now, let's check the ordered pair (2,0)(2, 0)(2,0). We substitute x=2x = 2x=2 and y=0y = 0y=0 into the first equation 4x−5y=84x - 5y = 84x−5y=8.4(2)−5(0)=84(2) - 5(0) = 84(2)−5(0)=8

STEP 17

Calculate the left-hand side of the first equation with the substituted values.8−0=88 - 0 = 88−0=8

STEP 18

Check if the left-hand side equals the right-hand side.8=88 = 88=8The left-hand side equals the right-hand side, so the ordered pair (2,0)(2, 0)(2,0) satisfies the first equation.

STEP 19

Now, substitute x=2x = 2x=2 and y=0y = 0y=0 into the second equation −3x+2y=1-3x + 2y = 1−3x+2y=1.−3(2)+2(0)=1-3(2) + 2(0) = 1−3(2)+2(0)=1

STEP 20

Calculate the left-hand side of the second equation with the substituted values.−6+0=1-6 + 0 = 1−6+0=1

STEP 21

Check if the left-hand side equals the right-hand side.−6≠1-6 \neq 1−6=1Since the left-hand side does not equal the right-hand side, the ordered pair (2,0)(2, 0)(2,0) is not a solution to the second equation.

STEP 22

We can conclude that the ordered pair (2,0)(2, 0)(2,0) is not a solution to the system of equations. We mark "No" in the table for this ordered pair.

STEP 23

Finally, let's check the ordered pair (3,5)(3, 5)(3,5). We substitute x=3x = 3x=3 and y=5y = 5y=5 into the first equation 4x−5y=84x - 5y = 84x−5y=8.4(3)−5(5)=84(3) - 5(5) = 84(3)−5(5)=8

STEP 24

Calculate the left-hand side of the first equation with the substituted values.12−25=812 - 25 = 812−25=8

STEP 25

Check if the left-hand side equals the right-hand side.−13≠8-13 \neq 8−13=8Since the left-hand side does not equal the right-hand side, the ordered pair (3,5)(3, 5)(3,5) is not a solution to the first equation.

STEP 26

We will now substitute x=3x = 3x=3 and y=5y = 5y=5 into the second equation −3x+2y=1-3x + 2y = 1−3x+2y=1.−3(3)+2(5)=1-3(3) + 2(5) = 1−3(3)+2(5)=1

STEP 27

Calculate the left-hand side of the second equation with the substituted values.−9+10=1-9 + 10 = 1−9+10=1

STEP 28

Check if the left-hand side equals the right-hand side.1=11 = 11=1The left-hand side equals the right-hand side, but since the ordered pair (3,5)(3, 5)(3,5) did not satisfy the first equation, it is not a solution to the system.

STEP 29

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Determine if each ordered pair $(x, y)$ is a solution to the system of linear equations $\begin{cases} 4x - 5y = 8\\ -3x + 2y = 1 \end{cases}$ .

Let's start with the first ordered pair $(-6, 2)$ . We will substitute $x = -6$ and $y = 2$ into the first equation $4x - 5y = 8$ .
$4(-6) - 5(2) = 8$

Calculate the left-hand side of the first equation with the substituted values.
$-24 - 10 = 8$

Check if the left-hand side equals the right-hand side.
$-34 \neq 8$
Since the left-hand side does not equal the right-hand side, the ordered pair $(-6, 2)$ is not a solution to the first equation.

We will now substitute $x = -6$ and $y = 2$ into the second equation $-3x + 2y = 1$ .
$-3(-6) + 2(2) = 1$

Calculate the left-hand side of the second equation with the substituted values.
$18 + 4 = 1$

Check if the left-hand side equals the right-hand side.
$22 \neq 1$
Since the left-hand side does not equal the right-hand side, the ordered pair $(-6, 2)$ is not a solution to the second equation either.

We can conclude that the ordered pair $(-6, 2)$ is not a solution to the system of equations. We mark "No" in the table for this ordered pair.

Next, let's check the ordered pair $(-3, -4)$ . We substitute $x = -3$ and $y = -4$ into the first equation $4x - 5y = 8$ .
$4(-3) - 5(-4) = 8$

Calculate the left-hand side of the first equation with the substituted values.
$-12 + 20 = 8$

Check if the left-hand side equals the right-hand side.
$8 = 8$
The left-hand side equals the right-hand side, so the ordered pair $(-3, -4)$ satisfies the first equation.

Now, substitute $x = -3$ and $y = -4$ into the second equation $-3x + 2y = 1$ .
$-3(-3) + 2(-4) = 1$

Calculate the left-hand side of the second equation with the substituted values.
$9 - 8 = 1$

Check if the left-hand side equals the right-hand side.
$1 = 1$
The left-hand side equals the right-hand side, so the ordered pair $(-3, -4)$ also satisfies the second equation.

We can conclude that the ordered pair $(-3, -4)$ is a solution to the system of equations. We mark "Yes" in the table for this ordered pair.

Now, let's check the ordered pair $(2, 0)$ . We substitute $x = 2$ and $y = 0$ into the first equation $4x - 5y = 8$ .
$4(2) - 5(0) = 8$

Calculate the left-hand side of the first equation with the substituted values.
$8 - 0 = 8$

Check if the left-hand side equals the right-hand side.
$8 = 8$
The left-hand side equals the right-hand side, so the ordered pair $(2, 0)$ satisfies the first equation.

Now, substitute $x = 2$ and $y = 0$ into the second equation $-3x + 2y = 1$ .
$-3(2) + 2(0) = 1$

Calculate the left-hand side of the second equation with the substituted values.
$-6 + 0 = 1$

Check if the left-hand side equals the right-hand side.
$-6 \neq 1$
Since the left-hand side does not equal the right-hand side, the ordered pair $(2, 0)$ is not a solution to the second equation.

We can conclude that the ordered pair $(2, 0)$ is not a solution to the system of equations. We mark "No" in the table for this ordered pair.

Finally, let's check the ordered pair $(3, 5)$ . We substitute $x = 3$ and $y = 5$ into the first equation $4x - 5y = 8$ .
$4(3) - 5(5) = 8$

Calculate the left-hand side of the first equation with the substituted values.
$12 - 25 = 8$

Check if the left-hand side equals the right-hand side.
$-13 \neq 8$
Since the left-hand side does not equal the right-hand side, the ordered pair $(3, 5)$ is not a solution to the first equation.

We will now substitute $x = 3$ and $y = 5$ into the second equation $-3x + 2y = 1$ .
$-3(3) + 2(5) = 1$

Calculate the left-hand side of the second equation with the substituted values.
$-9 + 10 = 1$

Check if the left-hand side equals the right-hand side.
$1 = 1$
The left-hand side equals the right-hand side, but since the ordered pair $(3, 5)$ did not satisfy the first equation, it is not a solution to the system.