5. Solve the following pair of linear equations by cross- multiplication. 2x + 5y - 52 = 0 and 3x - 4y + 14 = 0

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To solve the given pair of linear equations using the cross-multiplication method, follow these steps:

### Step 1: Write the equations in standard form
The given equations are:

\[
2x + 5y - 52 = 0
\]
\[
3x - 4y + 14 = 0
\]

These can be written as:

\[
2x + 5y = 52 \quad \text{(Equation 1)}
\]
\[
3x - 4y = -14 \quad \text{(Equation 2)}
\]

### Step 2: Arrange coefficients
Let's write the coefficients of the equations in the following format:

For Equation 1: \( a_1 = 2 \), \( b_1 = 5 \), \( c_1 = 52 \)  
For Equation 2: \( a_2 = 3 \), \( b_2 = -4 \), \( c_2 = -14 \)

### Step 3: Apply the cross-multiplication formula
The cross-multiplication formula is as follows:

\[
\frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{1}{a_1b_2 - a_2b_1}
\]

Substituting the values:

\[
\frac{x}{(5)(-14) - (-4)(52)} = \frac{y}{(52)(3) - (-14)(2)} = \frac{1}{(2)(-4) - (3)(5)}
\]

### Step 4: Simplify each expression
Now, simplify each part:

1. For \( x \):
   \[
   \frac{x}{-70 + 208} = \frac{x}{138}
   \]

2. For \( y \):
   \[
   \frac{y}{156 + 28} = \frac{y}{184}
   \]

3. For the constant \( 1 \):
   \[
   \frac{1}{-8 - 15} = \frac{1}{-23}
   \]

### Step 5: Solve for \( x \) and \( y \)
Using the cross-multiplication method:

1. Solve for \( x \):
   \[
   \frac{x}{138} = \frac{1}{-23} \implies x = \frac{138}{-23} = -6
   \]

2. Solve for \( y \):
   \[
   \frac{y}{184} = \frac{1}{-23} \implies y = \frac{184}{-23} = -8
   \]

### Solution:
The solution to the given pair of equations is:

\[
x = -6, \quad y = -8
\]

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Do you want details or have any questions? Here are some related questions:

1. What are the steps to solve a pair of linear equations using the substitution method?
2. How do you determine if two linear equations have a unique solution, no solution, or infinitely many solutions?
3. Can you solve a pair of linear equations using the elimination method?
4. How do you graphically represent the solution of a pair of linear equations?
5. What is the difference between consistent and inconsistent systems of linear equations?

**Tip:** Always check your solution by substituting the values of \( x \) and \( y \) back into the original equations to verify correctness.

Learn how to solve a pair of linear equations using the cross-multiplication method with a step-by-step solution. This example covers key concepts such as linear equations and the cross-multiplication formula. Suitable for Grades 9-12 students.

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