How to Solve the System Algebraically: 7x + 2y = 4 and 5y = 3x + 10?

To solve the system of equations algebraically, we have the following two equations:

• Equation 1: 7x + 2y = 4
• Equation 2: 5y = 3x + 10

First, let’s rearrange Equation 2 to express y in terms of x. We can do this by isolating y:

5y = 3x + 10

Dividing both sides by 5, we get:

y = (3/5)x + 2

Now that we have y in terms of x, we can substitute this expression into Equation 1:

7x + 2((3/5)x + 2) = 4

This simplifies to:

7x + (6/5)x + 4 = 4

Next, we can combine the x terms. To do this, let’s convert 7x into a fraction with a common denominator of 5:

(35/5)x + (6/5)x + 4 = 4

Now we can combine the fractions:

(41/5)x + 4 = 4

Next, we isolate the term involving x:

(41/5)x = 4 - 4

(41/5)x = 0

This means:

x = 0

Now that we have the value of x, let’s plug it back into our equation for y:

y = (3/5)(0) + 2

y = 2

So the solution to the system of equations is:

• x = 0
• y = 2

To check our solution, we can substitute x and y back into both original equations:

• 7(0) + 2(2) = 4 ⇒ 0 + 4 = 4 (True)
• 5(2) = 3(0) + 10 ⇒ 10 = 0 + 10 (True)

Both equations are satisfied, confirming our solution is correct: (x, y) = (0, 2).