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This graph shows the lines for these equations:

3y=3x + 6

y = x + 2

How many solutions are there for this system of equations?

3 answers

To determine how many solutions there are for the system of equations represented by the lines \(3y = 3x + 6\) and \(y = x + 2\), we can start by rewriting the first equation in the slope-intercept form (y = mx + b).

  1. Start with the equation \(3y = 3x + 6\).

  2. Divide every term by 3 to solve for \(y\):

    \[ y = x + 2 \]

Now we have both equations for the lines:

  1. \(y = x + 2\) (from the first equation)
  2. \(y = x + 2\) (the second equation)

Both equations are the same: \(y = x + 2\).

Since these two lines are identical, they overlap completely. This means that there are infinitely many solutions to this system of equations, as every point on the line is a solution.

Conclusion: There are infinitely many solutions.

The table shows a linear relationship between the variables x and y.
x y
5 16.5
7 21.5
11 31.5
What are the slope and y-intercept of the linear relationship?

To find the slope and y-intercept of the linear relationship between \(x\) and \(y\), we can use the points provided in the table:

  • (5, 16.5)
  • (7, 21.5)
  • (11, 31.5)

Step 1: Calculate the slope (m)

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Let's use the points (5, 16.5) and (7, 21.5):

  1. \(x_1 = 5\), \(y_1 = 16.5\)
  2. \(x_2 = 7\), \(y_2 = 21.5\)

Substituting these values into the slope formula:

\[ m = \frac{21.5 - 16.5}{7 - 5} = \frac{5}{2} = 2.5 \]

Step 2: Calculate the y-intercept (b)

We can use the slope-point form of the equation of a line:

\[ y = mx + b \]

We already know the slope \(m = 2.5\). We'll use one of the points to solve for \(b\). Let's use the point (5, 16.5):

\[ 16.5 = 2.5(5) + b \]

Calculating \(2.5(5)\):

\[ 16.5 = 12.5 + b \]

Now isolate \(b\):

\[ b = 16.5 - 12.5 = 4 \]

Conclusion

The slope of the linear relationship is \(m = 2.5\) and the y-intercept is \(b = 4\).

So, the final results are:

  • Slope (m): 2.5
  • Y-intercept (b): 4

The equation of the line is:

\[ y = 2.5x + 4 \]