Question Determine if ordered pairs satisfy a system of two linear equations: $y = \frac{1}{3}x - 2, y = \frac{2}{6}x - \frac{12}{6}$ . Complete a table and find the number of solutions.

Studdy Solution

STEP 1

Assumptions
1. We have two equations in the system: - $y = \frac{1}{3}x - 2$ - $y = \frac{2}{6}x - \frac{12}{6}$
2. We need to check if each ordered pair $(x, y)$ satisfies both equations.
3. The ordered pairs to be checked are $(9, 1)$ , $(18, 4)$ , and $(-6, -4)$ .

STEP 2

Simplify the second equation to match the format of the first equation.
$y = \frac{2}{6}x - \frac{12}{6}$

STEP 3

Reduce the fractions in the second equation.
$y = \frac{1}{3}x - 2$

STEP 4

Notice that both equations are now identical, which means they represent the same line. Therefore, any solution that satisfies one equation will satisfy the other.

STEP 5

Substitute the x-value of the first ordered pair $(9, 1)$ into the equation to check if the y-value matches.
$y = \frac{1}{3}(9) - 2$

STEP 6

Calculate the y-value for the first ordered pair.
$y = 3 - 2 = 1$

STEP 7

Since the calculated y-value matches the y-value of the ordered pair $(9, 1)$ , this ordered pair satisfies the system of equations.

STEP 8

Substitute the x-value of the second ordered pair $(18, 4)$ into the equation to check if the y-value matches.
$y = \frac{1}{3}(18) - 2$

STEP 9

Calculate the y-value for the second ordered pair.
$y = 6 - 2 = 4$

STEP 10

Since the calculated y-value matches the y-value of the ordered pair $(18, 4)$ , this ordered pair satisfies the system of equations.

STEP 11

Substitute the x-value of the third ordered pair $(-6, -4)$ into the equation to check if the y-value matches.
$y = \frac{1}{3}(-6) - 2$

STEP 12

Calculate the y-value for the third ordered pair.
$y = -2 - 2 = -4$

STEP 13

Since the calculated y-value matches the y-value of the ordered pair $(-6, -4)$ , this ordered pair satisfies the system of equations.

STEP 14

Based on the results from the table, we can conclude that all the given ordered pairs satisfy the system of equations.

STEP 15

Since both equations are identical after simplification, they represent the same line. Therefore, the system of equations has infinitely many solutions.
The completed table is as follows: \begin{tabular}{|c|c|} \hline Ordered Pair & Satisfies the System \\ \hline $(9,1)$ & Yes \\ \hline $(18,4)$ & Yes \\ \hline $(-6,-4)$ & Yes \\ \hline \end{tabular}
Part B: Based on the results from the table, this system of equations has infinitely many solutions.

Was this helpful?

Question Determine if ordered pairs satisfy a system of two linear equations: $y = \frac{1}{3}x - 2, y = \frac{2}{6}x - \frac{12}{6}$ . Complete a table and find the number of solutions.

Studdy Solution

STEP 1

Assumptions
1. We have two equations in the system: - $y = \frac{1}{3}x - 2$ - $y = \frac{2}{6}x - \frac{12}{6}$
2. We need to check if each ordered pair $(x, y)$ satisfies both equations.
3. The ordered pairs to be checked are $(9, 1)$ , $(18, 4)$ , and $(-6, -4)$ .

STEP 2

Simplify the second equation to match the format of the first equation.
$y = \frac{2}{6}x - \frac{12}{6}$

STEP 3

Reduce the fractions in the second equation.
$y = \frac{1}{3}x - 2$

STEP 4

Notice that both equations are now identical, which means they represent the same line. Therefore, any solution that satisfies one equation will satisfy the other.

STEP 5

Substitute the x-value of the first ordered pair $(9, 1)$ into the equation to check if the y-value matches.
$y = \frac{1}{3}(9) - 2$

STEP 6

Calculate the y-value for the first ordered pair.
$y = 3 - 2 = 1$

STEP 7

Since the calculated y-value matches the y-value of the ordered pair $(9, 1)$ , this ordered pair satisfies the system of equations.

STEP 8

Substitute the x-value of the second ordered pair $(18, 4)$ into the equation to check if the y-value matches.
$y = \frac{1}{3}(18) - 2$

STEP 9

Calculate the y-value for the second ordered pair.
$y = 6 - 2 = 4$

STEP 10

Since the calculated y-value matches the y-value of the ordered pair $(18, 4)$ , this ordered pair satisfies the system of equations.

STEP 11

Substitute the x-value of the third ordered pair $(-6, -4)$ into the equation to check if the y-value matches.
$y = \frac{1}{3}(-6) - 2$

STEP 12

Calculate the y-value for the third ordered pair.
$y = -2 - 2 = -4$

STEP 13

Since the calculated y-value matches the y-value of the ordered pair $(-6, -4)$ , this ordered pair satisfies the system of equations.

STEP 14

Based on the results from the table, we can conclude that all the given ordered pairs satisfy the system of equations.

STEP 15

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question Determine if ordered pairs satisfy a system of two linear equations: y=13x−2,y=26x−126y = \frac{1}{3}x - 2, y = \frac{2}{6}x - \frac{12}{6}y=31​x−2,y=62​x−612​. Complete a table and find the number of solutions.

Studdy Solution

STEP 1

STEP 2

Simplify the second equation to match the format of the first equation.y=26x−126 y = \frac{2}{6}x - \frac{12}{6} y=62​x−612​

STEP 3

Reduce the fractions in the second equation.y=13x−2 y = \frac{1}{3}x - 2 y=31​x−2

STEP 4

Notice that both equations are now identical, which means they represent the same line. Therefore, any solution that satisfies one equation will satisfy the other.

STEP 5

Substitute the x-value of the first ordered pair (9,1)(9, 1)(9,1) into the equation to check if the y-value matches.y=13(9)−2 y = \frac{1}{3}(9) - 2 y=31​(9)−2

STEP 6

Calculate the y-value for the first ordered pair.y=3−2=1 y = 3 - 2 = 1 y=3−2=1

STEP 7

Since the calculated y-value matches the y-value of the ordered pair (9,1)(9, 1)(9,1), this ordered pair satisfies the system of equations.

STEP 8

Substitute the x-value of the second ordered pair (18,4)(18, 4)(18,4) into the equation to check if the y-value matches.y=13(18)−2 y = \frac{1}{3}(18) - 2 y=31​(18)−2

STEP 9

Calculate the y-value for the second ordered pair.y=6−2=4 y = 6 - 2 = 4 y=6−2=4

STEP 10

Since the calculated y-value matches the y-value of the ordered pair (18,4)(18, 4)(18,4), this ordered pair satisfies the system of equations.

STEP 11

Substitute the x-value of the third ordered pair (−6,−4)(-6, -4)(−6,−4) into the equation to check if the y-value matches.y=13(−6)−2 y = \frac{1}{3}(-6) - 2 y=31​(−6)−2

STEP 12

Calculate the y-value for the third ordered pair.y=−2−2=−4 y = -2 - 2 = -4 y=−2−2=−4

STEP 13

Since the calculated y-value matches the y-value of the ordered pair (−6,−4)(-6, -4)(−6,−4), this ordered pair satisfies the system of equations.

STEP 14

Based on the results from the table, we can conclude that all the given ordered pairs satisfy the system of equations.

STEP 15

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question Determine if ordered pairs satisfy a system of two linear equations: $y = \frac{1}{3}x - 2, y = \frac{2}{6}x - \frac{12}{6}$ . Complete a table and find the number of solutions.

Simplify the second equation to match the format of the first equation.
$y = \frac{2}{6}x - \frac{12}{6}$

Reduce the fractions in the second equation.
$y = \frac{1}{3}x - 2$

Substitute the x-value of the first ordered pair $(9, 1)$ into the equation to check if the y-value matches.
$y = \frac{1}{3}(9) - 2$

Calculate the y-value for the first ordered pair.
$y = 3 - 2 = 1$

Since the calculated y-value matches the y-value of the ordered pair $(9, 1)$ , this ordered pair satisfies the system of equations.

Substitute the x-value of the second ordered pair $(18, 4)$ into the equation to check if the y-value matches.
$y = \frac{1}{3}(18) - 2$

Calculate the y-value for the second ordered pair.
$y = 6 - 2 = 4$

Since the calculated y-value matches the y-value of the ordered pair $(18, 4)$ , this ordered pair satisfies the system of equations.

Substitute the x-value of the third ordered pair $(-6, -4)$ into the equation to check if the y-value matches.
$y = \frac{1}{3}(-6) - 2$

Calculate the y-value for the third ordered pair.
$y = -2 - 2 = -4$

Since the calculated y-value matches the y-value of the ordered pair $(-6, -4)$ , this ordered pair satisfies the system of equations.