What is the solution to the system of linear equations represented by the equations 3y + x = 6 and y + x = 3?

To solve the system of linear equations given by:

• Equation 1: 3y + x = 6
• Equation 2: y + x = 3

We can use the substitution or elimination method. Here, let’s use the substitution method for clarity.

Step 1: Solve one equation for one variable

We can start with Equation 2:

y + x = 3

To express y in terms of x, we rearrange the equation:

y = 3 - x

Step 2: Substitute into the other equation

Now, we substitute this expression for y into Equation 1:

3(3 - x) + x = 6

Distributing the terms:

9 - 3x + x = 6

This simplifies to:

9 - 2x = 6

Step 3: Solve for x

Now, isolate x:

-2x = 6 - 9

-2x = -3

x = rac{3}{2} = 1.5

Step 4: Substitute back to find y

Now that we have the value of x, we can substitute it back into our expression for y:

y = 3 - 1.5

y = 1.5

Step 5: Write the solution

Thus, the solution to the system of equations is:

• x = 1.5
• y = 1.5

In coordinate form, the solution can be expressed as:

(x, y) = (1.5, 1.5)

To summarize, the solution to the system of equations 3y + x = 6 and y + x = 3 is:

(1.5, 1.5)