To solve the system of equations for the intersection of the two lines:

1. Equations given:
   • $3x + 4y = 10$
   • $5x - 6y = 23$

We aim to solve this system for $x$ and $y$.

Step 1: Solve one equation for one variable

Let's solve the first equation for $x$: $3x + 4y = 10 \quad \Rightarrow \quad x = \frac{10 - 4y}{3}.$

Step 2: Substitute into the second equation

Substitute $x = \frac{10 - 4y}{3}$ into the second equation $5x - 6y = 23$: $5\left(\frac{10 - 4y}{3}\right) - 6y = 23.$

Simplify: $\frac{50 - 20y}{3} - 6y = 23.$

Multiply through by 3 to eliminate the fraction: $50 - 20y - 18y = 69.$

Simplify: $50 - 38y = 69 \quad \Rightarrow \quad -38y = 19 \quad \Rightarrow \quad y = -\frac{19}{38} = -\frac{1}{2}.$

Step 3: Solve for $x$

Substitute $y = -\frac{1}{2}$ into $x = \frac{10 - 4y}{3}$: $x = \frac{10 - 4(-\frac{1}{2})}{3} = \frac{10 + 2}{3} = \frac{12}{3} = 4.$

Final Answer:

The point of intersection is: $(x, y) = \left( 4, -\frac{1}{2} \right).$

Let me know if you’d like a detailed explanation of any step!

Related Questions:

1. How do you solve systems of equations using elimination?
2. How do you interpret the solution of a system of equations geometrically?
3. What happens if two lines are parallel?
4. How do you solve systems of equations using matrices?
5. How does substitution compare to elimination in terms of efficiency?

Tip: Always double-check your solution by substituting the values of $x$ and $y$ into both original equations!