Question

5. Andre does not understand why a solution to the equation 3 - x = 4 mustalso be a solution to the equation 12 = 9 - 3x.Write a convincing explanationas to why this is true.

Answer 1

Both equations, 3 - x = 4 and 12 = 9 - 3x, yield the same solution x = -1 when solved separately. This is because the equations are related by multiplication by -3, which allows any solution of one equation to also be a solution of the other, demonstrating the universal rules of algebra.

Step-by-step explanation:

To demonstrate why a solution to the equation 3 - x = 4 must also be a solution to the equation 12 = 9 - 3x, let's solve both equations and observe their relationship.

First, let's solve the equation 3 - x = 4:

1. Subtract 3 from both sides to isolate the variable x on one side: 3 - x - 3 = 4 - 3, which simplifies to -x = 1.
2. Multiply both sides by -1 to get x = -1.

Now, let's solve 12 = 9 - 3x:

1. Subtract 9 from both sides: 12 - 9 = 9 - 9 - 3x, which simplifies to 3 = -3x.
2. Divide both sides by -3 to solve for x: 3 / (-3) = (-3x) / (-3), which simplifies to x = -1.

As we can see, both equations yield the same solution, x = -1. This happens because the operations we perform to isolate x in both cases are based on the fundamental rules of algebra that apply universally. These rules include operations such as adding or subtracting the same value from both sides of an equation and multiplying or dividing both sides of an equation by a non-zero number. The equations 3 - x = 4 and 12 = 9 - 3x are related by multiplication by -3, which transforms one into the other, ensuring that any solution for one equation must also satisfy the other.

Answer 2

The first equation is 3 - x = 4 while the second equation is 12 = 9 - 3x.

By careful observation, we can see that both sides of the first equation has been multiplied by a constant factor 3 to form the second equation. Because this is an eqaution and both sides of the equation has been multiplied by thesame factor, the solution of the equation will remain the same because a balance is maintained.

Remember that to maintain a balance to an equation, whatever is done to one side of the equation must be done to the other side of the equation.

Thus,

3 + x = 4

After multiplying through by 3, we have

3( 3 + x ) = 3(4)

9 + 3x = 12

Since 3 is multiplying both sides of the equation, The solution to the equations remain equal