Ask a New Question
Menu

Determine which three out of the six points shown below are a solution to the equation 2, x, plus, y, equals, 6, .2x+y=6. Plot the three points that are in the solution set, draw a line through the three points and then answer the questions below.

left parenthesis, 6, comma, 2, right parenthesis(6,2), left parenthesis, minus, 2, comma, 2, right parenthesis(−2,2), left parenthesis, 4, comma, minus, 2, right parenthesis(4,−2), left parenthesis, 0, comma, 6, right parenthesis(0,6), left parenthesis, 5, comma, minus, 8, right parenthesis(5,−8), left parenthesis, 6, comma, minus, 6, right parenthesis(6,−6)

3 answers

To determine which points satisfy the equation \( 2x + y = 6 \), we can substitute the \( x \) and \( y \) values from each point into the equation and check if it holds true.

  1. For the point \( (6, 2) \): \[ 2(6) + 2 = 12 + 2 = 14 \quad \text{(not a solution)} \]

  2. For the point \( (-2, 2) \): \[ 2(-2) + 2 = -4 + 2 = -2 \quad \text{(not a solution)} \]

  3. For the point \( (4, -2) \): \[ 2(4) + (-2) = 8 - 2 = 6 \quad \text{(solution)} \]

  4. For the point \( (0, 6) \): \[ 2(0) + 6 = 0 + 6 = 6 \quad \text{(solution)} \]

  5. For the point \( (5, -8) \): \[ 2(5) + (-8) = 10 - 8 = 2 \quad \text{(not a solution)} \]

  6. For the point \( (6, -6) \): \[ 2(6) + (-6) = 12 - 6 = 6 \quad \text{(solution)} \]

The points that are solutions to the equation \( 2x + y = 6 \) are \( (4, -2) \), \( (0, 6) \), and \( (6, -6) \).

Plotting the Points

  • (4, -2)
  • (0, 6)
  • (6, -6)

Equation of the Line

To draw the line through these points, we can express the equation in slope-intercept form \( y = mx + b \). The equation \( 2x + y = 6 \) can be rewritten as: \[ y = -2x + 6 \] Where the slope \( m = -2 \) and the y-intercept \( b = 6 \).

Summary

The three points that are solutions to the equation \( 2x + y = 6 \) are:

  1. \( (4, -2) \)
  2. \( (0, 6) \)
  3. \( (6, -6) \)

After plotting these points and drawing the line, you'll clearly see they all lie on the line \( y = -2x + 6 \).

Come up with two additional solutions now that you have graphed the line. Justify your solution numerically. At least one of your points must include decimals or fractions.

New Solution #1: left parenthesis(
,
right parenthesis) is a solution because when you plug it into the original equation 2, left parenthesis2(
right parenthesis, plus, left parenthesis)+(
right parenthesis, equals, 6)=6, the equation simplifies to
equals, 6=6.
New Solution #2: left parenthesis(
,
right parenthesis) is a solution because when you plug it into the original equation 2, left parenthesis2(
right parenthesis, plus, left parenthesis)+(
right parenthesis, equals, 6)=6, the equation simplifies to
equals, 6=6.
Submit Answer

Here are two additional solutions to the equation \( 2x + y = 6 \):

New Solution #1

Let's choose \( x = 1 \):

  1. Substitute \( x = 1 \) into the equation to find \( y \): \[ 2(1) + y = 6 \] \[ 2 + y = 6 \] \[ y = 6 - 2 = 4 \] Therefore, New Solution #1 is \( (1, 4) \) because when you plug it into the original equation \( 2(1) + 4 = 6 \), the equation simplifies to \( 2 + 4 = 6 \).

New Solution #2

Let's choose \( x = 2.5 \):

  1. Substitute \( x = 2.5 \) into the equation to find \( y \): \[ 2(2.5) + y = 6 \] \[ 5 + y = 6 \] \[ y = 6 - 5 = 1 \] Therefore, New Solution #2 is \( (2.5, 1) \) because when you plug it into the original equation \( 2(2.5) + 1 = 6 \), the equation simplifies to \( 5 + 1 = 6 \).

Summary of New Solutions

  • New Solution #1: \( \left(1, 4\right) \)
  • New Solution #2: \( \left(2.5, 1\right) \)

Both points satisfy the equation \( 2x + y = 6 \).