Answered

Set up a table of $x, y$ pairs that fits the following linear function. For your convenience, $x$ values have been selected. Just calculate the corresponding $y$-values, plot the points on a Cartesian coordinate system, and draw the line.



$2y + 2x = 6$




$x$		$y$

0		A

1		B

2		C

3		D






Choose the correct sequence of $y$-values (A, B, C, D) that would fit the table shown. Graph the line on your own paper.


A3, 2, 1, 0
B3, 1, -1, -3
C0, 1, 2, 3
D$1, 2, 3, 4$

Verified Answer :

  • Isolate the variable \(y\) in the linear equation \(2y + 2x = 6\) to obtain \(y = 3 - x\).
  • Calculate the \(y\)-values for \(x = 0, 1, 2, 3\) by substituting them into the equation \(y = 3 - x\).
  • Determine the sequence of \(y\)-values is \((3, 2, 1, 0)\).
  • Conclude that the correct option is \(\boxed{a}\).

Explanation

  1. Isolating the Variable y

The given linear equation is \(2y + 2x = 6\). To find the corresponding \(y\)-values for the specified \(x\)-values, we first isolate \(y\) in terms of \(x\). We can do this by subtracting \(2x\) from both sides of the equation:
\[2y = 6 - 2x.\]
Next, divide both sides by \(2\) to solve for \(y\):
\[y = \frac{6 - 2x}{2} = 3 - x.\]
This provides us with a clear linear relationship where each \(y\)-value is calculated by subtracting the corresponding \(x\)-value from \(3\).

  1. Calculating y-values for the given x-values

Now, we substitute the given \(x\)-values into the equation \(y = 3 - x\) to find the corresponding \(y\)-values:

  • For \(x = 0\): \(A = 3 - 0 = 3\).
  • For \(x = 1\): \(B = 3 - 1 = 2\).
  • For \(x = 2\): \(C = 3 - 2 = 1\).
  • For \(x = 3\): \(D = 3 - 3 = 0\).

This gives us the sequence of \(y\)-values \((3, 2, 1, 0)\).

  1. Comparing with Multiple-Choice Options

Comparing our calculated sequence \((3, 2, 1, 0)\) with the given multiple-choice options:

  • a. \(3, 2, 1, 0\)
  • b. \(3, 1, -1, -3\)
  • c. \(0, 1, 2, 3\)
  • d. \(1, 2, 3, 4\)

The sequence \((3, 2, 1, 0)\) exactly matches option a. Thus, the correct sequence is a.

Examples

Linear relationships are fundamental in many daily activities, such as calculating how long a battery will last based on usage. If you have a device that starts with a certain amount of power (like 3 hours of charge) and it consumes power at a constant rate, the remaining battery life can be modeled by a linear function like \(y = 3 - x\), where \(x\) is the number of hours used and \(y\) is the remaining hours. This helps in planning your day to ensure your devices don't run out of power when you need them most.

Related Questions