Find \(\alpha\) and \(\beta\)

📝 Question

Given:

\[ g=\{(1,1),(2,3),(3,5),(4,7)\} \]

and \(g(x)=\alpha x+\beta\). Find \(\alpha\) and \(\beta\).

---

✅ Solution

🔹 Step 1: Form equations

Using \((1,1)\):

\[ \alpha(1)+\beta=1 \]

\[ \alpha+\beta=1 \quad …(1) \]

Using \((2,3)\):

\[ 2\alpha+\beta=3 \quad …(2) \] —

🔹 Step 2: Solve equations

Subtract (1) from (2):

:contentReference[oaicite:0]{index=0}

Substitute into (1):

\[ 2+\beta=1 \Rightarrow \beta=-1 \] —

🎯 Final Answer

\[ \boxed{\alpha=2,\quad \beta=-1} \]

---

🚀 Exam Shortcut

• Take any two points
• Form linear equations
• Solve quickly by subtraction
• Check with remaining points