To determine the graph that shows the solution for the equations \(y = x + 2\) and \(54 = 3x + 2y\), we first need to rewrite both equations in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
-
The first equation is already in the appropriate form: \[ y = x + 2 \] This means the line has a slope of 1 and a y-intercept at (0, 2).
-
For the second equation \(54 = 3x + 2y\), we can isolate \(y\): \[ 2y = 54 - 3x \] \[ y = -\frac{3}{2}x + 27 \] Here, the slope is \(-\frac{3}{2}\) and the y-intercept is (0, 27).
Next, we can analyze both equations to find their intersection, as that point will be the solution to the system of equations.
To find the intersection of: \[ y = x + 2 \] and \[ y = -\frac{3}{2}x + 27 \]
We can set their right-hand sides equal to each other: \[ x + 2 = -\frac{3}{2}x + 27 \]
Now, solving for \(x\):
-
Combine like terms: \[ x + \frac{3}{2}x = 27 - 2 \] \[ \frac{5}{2}x = 25 \]
-
Multiply both sides by \(\frac{2}{5}\): \[ x = 10 \]
-
Substitute \(x = 10\) back into the first equation to find \(y\): \[ y = 10 + 2 = 12 \]
Thus, the solution to the system is \( (10, 12) \).
To represent this graphically, you would draw both lines:
- The line \(y = x + 2\) will have points like (0, 2), (1, 3), (2, 4), etc.
- The line \(y = -\frac{3}{2}x + 27\) will have points like (0, 27), (2, 24), (4, 21), etc.
The point of intersection at (10, 12) should appear on both lines in the graph you are considering. Thus, the correct graph should display both lines and show that they intersect at the point (10, 12).
1 answer