To find the intersection of the two boundaries represented by the equations \(y = -2x - 5\) and \(-2x + 4y = 12\), we first need to convert the second equation into slope-intercept form (if necessary) and then solve the two equations simultaneously.
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First Equation: \(y = -2x - 5\)
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Second Equation: \(-2x + 4y = 12\)
To convert this to slope-intercept form, we can solve for \(y\): \[ 4y = 2x + 12 \ y = \frac{1}{2}x + 3 \]
Now we have the two equations:
- \(y = -2x - 5\)
- \(y = \frac{1}{2}x + 3\)
Next, we can set these two equations equal to each other to find the intersection point: \[ -2x - 5 = \frac{1}{2}x + 3 \]
Now, let's eliminate the fraction by multiplying everything by 2: \[ -4x - 10 = x + 6 \]
Now, combine like terms: \[ -4x - x = 6 + 10 \ -5x = 16 \ x = -\frac{16}{5} \ x = -3.2 \]
Now, substitute \(x = -3.2\) back into one of the original equations to find \(y\). We'll use the first equation: \[ y = -2(-3.2) - 5 \ y = 6.4 - 5 \ y = 1.4 \]
Thus, the intersection point is \((-3.2, 1.4)\).
Now, you should check the provided coordinate grids (AA, BB, CC, DD) to identify which graph accurately depicts the two equations and the calculated intersection point of \((-3.2, 1.4)\).
I cannot view the grids directly, but you should look for:
- The first line \(y = -2x - 5\) which has a steep downward slope and intersects the y-axis at \(-5\).
- The second line \(y = \frac{1}{2}x + 3\) which has a gentle upward slope and intersects the y-axis at \(3\).
- The intersection point \((-3.2, 1.4)\) should be marked as the zoo's entrance.
Make sure to visually confirm which option aligns with these characteristics.