What is the solution to this system of linear equations 3x + 2y = 14 and 5x + y = 32?

To solve the system of equations:

• 1. 3x + 2y = 14
• 2. 5x + y = 32

We can use the method of substitution or elimination. Here, we will use substitution for clarity.

First, let’s solve the second equation for y:

y = 32 - 5x

Now, substitute this expression for y into the first equation:

3x + 2(32 - 5x) = 14

This simplifies to:

3x + 64 - 10x = 14

Combine like terms:

-7x + 64 = 14

Now, isolate x:

-7x = 14 - 64

-7x = -50

x = rac{50}{7}

Now that we have x, we can substitute back to find y. Using our expression for y:

y = 32 - 5(rac{50}{7})

Calculating that gives:

y = 32 - rac{250}{7}

Convert 32 to a fraction:

y = rac{224}{7} - rac{250}{7} = -rac{26}{7}

Therefore, the solution to the system of equations is:

• x = rac{50}{7}
• y = -rac{26}{7}

In decimal form, this is approximately:

• x ≈ 7.14
• y ≈ -3.71

This gives us the points where the two equations intersect on a graph.