Solve the following pair of linear equations by the substitution method.

• 3x/2 - 5y/3 = -2
• x/3 + y/2 = 13/6

To solve the given pair of linear equations using the substitution method, we start with the two equations:

Equations

• Equation 1: 3x/2 - 5y/3 = -2
• Equation 2: x/3 + y/2 = 13/6

Step 1: Solve one equation for a variable

Let's solve Equation 2 for y.

First, multiply the entire equation by 6 to eliminate the fractions:

2x + 3y = 13

Now, isolate y:

3y = 13 - 2x

y = (13 - 2x)/3

Step 2: Substitute into the other equation

Next, substitute this expression for y into Equation 1:

3x/2 - 5((13 - 2x)/3)/3 = -2

Now, simplify:

3x/2 - (65 - 10x)/9 = -2

Step 3: Clear the fractions

Multiply through by 18 (the least common multiple of 2 and 9) to eliminate the fractions:

27x - 2(65 - 10x) = -36

Distributing gives:

27x - 130 + 20x = -36

Step 4: Combine like terms

Combine the x terms:

47x - 130 = -36

Now, add 130 to both sides:

47x = 94

Step 5: Solve for x

Divide by 47:

x = 2

Step 6: Find y

Now substitute x = 2 back into the expression for y:

y = (13 - 2(2))/3

y = (13 - 4)/3

y = 9/3

y = 3

Final Solution

The solution to the system of equations is:

• x = 2
• y = 3