To solve the system of equations using the substitution method, we have the following equations:
- \( y = x^2 - 4x + 8 \)
- \( y = 3x - 4 \)
We can set the two equations equal to each other since both are equal to \( y \):
\[ x^2 - 4x + 8 = 3x - 4 \]
Now we can rearrange this equation to find the values of \( x \):
\[ x^2 - 4x - 3x + 8 + 4 = 0 \]
This simplifies to:
\[ x^2 - 7x + 12 = 0 \]
Next, we can factor this quadratic equation:
\[ (x - 3)(x - 4) = 0 \]
This gives us two possible values for \( x \):
\[ x = 3 \quad \text{or} \quad x = 4 \]
Now, we can find the corresponding \( y \) values for each \( x \) using the second equation \( y = 3x - 4 \):
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For \( x = 3 \): \[ y = 3(3) - 4 = 9 - 4 = 5 \quad \Rightarrow \quad (3, 5) \]
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For \( x = 4 \): \[ y = 3(4) - 4 = 12 - 4 = 8 \quad \Rightarrow \quad (4, 8) \]
Thus, the solutions to the system of equations are \( (3, 5) \) and \( (4, 8) \).
From the options given, the correct answer is:
(4, 8) and (3, 5)