Question

What is the x-coordinate value of the first point that satisfies the given simultaneous inequalities y≤3/5x−4 and y≤1/5x+1?

Answer 1

To find the x-coordinate of the first point that satisfies the given simultaneous inequalities \( y \leq \frac{3}{5}x - 4 \) and \( y \leq \frac{1}{5}x + 1 \), we can first determine the points at which the two boundary lines intersect by setting the equations equal to each other.

1. **Set the equations equal to find the intersection:**

\[
\frac{3}{5}x - 4 = \frac{1}{5}x + 1
\]

2. **Clear the fractions by multiplying through by 5:**

\[
3x - 20 = x + 5
\]

3. **Rearrange the equation:**

\[
3x - x = 20 + 5
\]
\[
2x = 25
\]

4. **Solve for x:**

\[
x = \frac{25}{2} = 12.5
\]

5. **Now, find the corresponding y-coordinate by substituting \( x = 12.5 \) into one of the equations. We'll use the first one:**

\[
y = \frac{3}{5}(12.5) - 4
\]
\[
y = \frac{3 \times 12.5}{5} - 4 = \frac{37.5}{5} - 4 = 7.5 - 4 = 3.5
\]

So the intersection point of the two lines is \( (12.5, 3.5) \).

6. **Since we want the x-coordinate of the first point that satisfies both inequalities, it occurs at their intersection.**

Thus, the x-coordinate value of the first point that satisfies the simultaneous inequalities is

\[
\boxed{12.5}.
\]