https://file.aiquickdraw.com/mathbot/file/1727384902703osm0bpti.jpg

Let's break down the steps based on the image provided:

### (a) Solve the first equation for \( x \)

The first equation is:
\[
5x - 6y = 1
\]
To solve for \( x \), we can isolate it on one side of the equation:

\[
5x = 6y + 1
\]

Now divide both sides by 5:

\[
x = \frac{6y + 1}{5}
\]

---

### (b) Substitute your result from part (a) into the second equation

The second equation is:
\[
15x - 18y = 3
\]

Substitute the value of \( x \) from part (a):
\[
15\left(\frac{6y + 1}{5}\right) - 18y = 3
\]

Simplify:
\[
15 \cdot \frac{6y + 1}{5} = 3(6y + 1)
\]
So the equation becomes:
\[
3(6y + 1) - 18y = 3
\]

Expanding:
\[
18y + 3 - 18y = 3
\]

Now simplify:
\[
3 = 3
\]

This shows that the equation is always true, regardless of the value of \( y \), meaning \( x \) and \( y \) are not uniquely determined by these equations.

---

### (c) How does this result tell us that every ordered pair \((x, y)\) satisfies the second equation?

Since the equation simplifies to a true statement \( 3 = 3 \), it means that the second equation doesn't impose any additional restrictions on \( x \) or \( y \). This implies that any pair of values \( (x, y) \) that satisfies the first equation will automatically satisfy the second equation as well.

---

Would you like further details on any of the steps? Here are five follow-up questions to deepen your understanding:

1. What happens if you try to solve the system graphically?
2. Why is it important to check if an equation is always true?
3. How do we interpret this solution in terms of the geometry of lines?
4. Can we express this system as a dependent system of equations?
5. What is the general form of solutions for this type of system?

**Tip:** When solving a system of equations, always check for consistency by substituting back into both equations.

In the Exercise 5.1.4, we found several ordered pairs that satisfy both of the equations: (1) 5x - 6y = 1, (2) 15x - 18y = 3. (a) Solve the first equation for x. (b) Substitute your result from part (a) into the second equation. After you simplify as much as possible, do you still have an x in your equation? Is the equation you have after simplifying true? (c) How does the result of part (b) tell us that every ordered pair (x, y) satisfies the second equation?

Learn how to solve the system of linear equations 5x - 6y = 1 and 15x - 18y = 3. This problem uses the substitution method to explore the consistency of the system and shows that every ordered pair (x, y) that satisfies the first equation also satisfies the second. Suitable for Grade 8 students.

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