Certainly! Let’s go through the problem step-by-step to solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Here are the equations given:
1. [tex]\( 3x + 4 = 9 \)[/tex]
2. [tex]\( y = 4 - 3x \)[/tex]
3. [tex]\( y = 9 - 3(2) \)[/tex]
### Step-by-Step Solution:
#### Step 1: Solve for [tex]\( x \)[/tex] in the first equation
Start with the equation:
[tex]\[ 3x + 4 = 9 \][/tex]
Subtract 4 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ 3x = 9 - 4 \][/tex]
[tex]\[ 3x = 5 \][/tex]
Divide both sides by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{5}{3} \][/tex]
So, the solution for [tex]\( x \)[/tex] is:
[tex]\[ x = \frac{5}{3} \][/tex]
#### Step 2: Calculate [tex]\( y \)[/tex] using the second equation and a given [tex]\( x \)[/tex]-value
We are given a specific value for [tex]\( x \)[/tex]:
[tex]\[ x = 2 \][/tex]
Use this [tex]\( x \)[/tex]-value in the second equation [tex]\( y = 4 - 3x \)[/tex]:
Substitute [tex]\( x = 2 \)[/tex] into the equation:
[tex]\[ y = 4 - 3(2) \][/tex]
[tex]\[ y = 4 - 6 \][/tex]
[tex]\[ y = -2 \][/tex]
So when [tex]\( x = 2 \)[/tex], the value of [tex]\( y \)[/tex] is:
[tex]\[ y = -2 \][/tex]
#### Step 3: Verify the value of [tex]\( y \)[/tex] using substitution
We also need to verify the value of [tex]\( y \)[/tex] with the third equation given explicitly for when [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 9 - 3(2) \][/tex]
Perform the calculation:
[tex]\[ y = 9 - 6 \][/tex]
[tex]\[ y = 3 \][/tex]
So substituting [tex]\( x = 2 \)[/tex] into the third equation gives:
[tex]\[ y = 3 \][/tex]
### Summary of Solutions
- The solution for [tex]\( x \)[/tex] from the first equation is [tex]\( x = \frac{5}{3} \)[/tex].
- By using [tex]\( x = 2 \)[/tex] in the second equation, the value of [tex]\( y \)[/tex] is [tex]\( y = -2 \)[/tex].
- Substituting [tex]\( x = 2 \)[/tex] into the third equation gives [tex]\( y = 3 \)[/tex].
The numerical results are:
- Solution for [tex]\( x \)[/tex] is [tex]\( \frac{5}{3} \)[/tex] (approximately 1.67)
- Value of [tex]\( y \)[/tex] when [tex]\( x = 2 \)[/tex] in the second equation is [tex]\( -2 \)[/tex]
- Value of [tex]\( y \)[/tex], verified with substitution in the third equation, is [tex]\( 3 \)[/tex]
I hope this detailed solution helps!
