5(a) (a) Using Cramer's rule to determine $$ x+y $$, if $$ \left[\begin{array}{cc}5 & -3 \ 4 & 2\end{array} ight]\left[\begin{array}{l}x \ y\end{array} ight]=\left[\begin{array}{c}-1 \ -8\end{array} ight] $$ (b) Given IS equation $$ 0.4 Y+100 I-255=0 $$ and the $$ L M $$ equation $$ 0.25 Y-200 \mathrm{I}-177=0 $$. Use matrix form to find the equilibrium level of income and rate of interest.

Question

5(a) (a) Using Cramer's rule to determine $$ x+y $$, if $$ \left[\begin{array}{cc}5 & -3 \ 4 & 2\end{array}ight]\left[\begin{array}{l}x \ y\end{array}ight]=\left[\begin{array}{c}-1 \ -8\end{array}ight] $$ (b) Given IS equation $$ 0.4 Y+100 I-255=0 $$ and the $$ L M $$ equation $$ 0.25 Y-200 \mathrm{I}-177=0 $$. Use matrix form to find the equilibrium level of income and rate of interest.

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Let's solve each part of the problem step by step.

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### **Problem 5(a): Using Cramer's Rule to Determine $$ x + y $$**

**Given:**
$$
\begin{bmatrix}
5 & -3 \
4 & 2
\end{bmatrix}
\begin{bmatrix}
x \
y
\end{bmatrix}
=
\begin{bmatrix}
-1 \
-8
\end{bmatrix}
$$

**System of Equations:**
$$
\begin{cases}
5x - 3y = -1 \quad \text{(1)} \
4x + 2y = -8 \quad \text{(2)}
\end{cases}
$$

**Step 1: Compute the Determinant (D) of the Coefficient Matrix**
$$
D = \begin{vmatrix}
5 & -3 \
4 & 2
\end{vmatrix} = (5)(2) - (-3)(4) = 10 + 12 = 22
$$

**Step 2: Compute Determinants for $$ x $$ and $$ y $$ (Dx and Dy)**
$$
D_x = \begin{vmatrix}
-1 & -3 \
-8 & 2
\end{vmatrix} = (-1)(2) - (-3)(-8) = -2 - 24 = -26
$$
$$
D_y = \begin{vmatrix}
5 & -1 \
4 & -8
\end{vmatrix} = (5)(-8) - (-1)(4) = -40 + 4 = -36
$$

**Step 3: Solve for $$ x $$ and $$ y $$ Using Cramer's Rule**
$$
x = \frac{D_x}{D} = \frac{-26}{22} = -\frac{13}{11}
$$
$$
y = \frac{D_y}{D} = \frac{-36}{22} = -\frac{18}{11}
$$

**Step 4: Determine $$ x + y $$**
$$
x + y = -\frac{13}{11} + (-\frac{18}{11}) = -\frac{31}{11}
$$

**Final Answer for Part (a):**
$$
x + y = -\frac{31}{11}
$$

---

### **Problem 5(b): Finding the Equilibrium Level of Income and Rate of Interest Using Matrix Methods**

**Given Equations:**
$$
\begin{cases}
0.4Y + 100I - 255 = 0 \quad \text{(IS Equation)} \
0.25Y - 200I - 177 = 0 \quad \text{(LM Equation)}
\end{cases}
$$

**Step 1: Rewrite the Equations in Standard Form**
$$
\begin{cases}
0.4Y + 100I = 255 \quad \text{(1)} \
0.25Y - 200I = 177 \quad \text{(2)}
\end{cases}
$$

**Step 2: Express in Matrix Form $$ AX = B $$**
$$
A = \begin{bmatrix}
0.4 & 100 \
0.25 & -200
\end{bmatrix}, \quad
X = \begin{bmatrix}
Y \
I
\end{bmatrix}, \quad
B = \begin{bmatrix}
255 \
177
\end{bmatrix}
$$

**Step 3: Compute the Determinant of Matrix $$ A $$**
$$
D = (0.4)(-200) - (100)(0.25) = -80 - 25 = -105
$$

**Step 4: Find the Inverse of Matrix $$ A $$**
$$
A^{-1} = \frac{1}{D} \begin{bmatrix}
-200 & -100 \
-0.25 & 0.4
\end{bmatrix} = \frac{1}{-105} \begin{bmatrix}
-200 & -100 \
-0.25 & 0.4
\end{bmatrix}
$$

**Step 5: Multiply $$ A^{-1} $$ with $$ B $$ to Find $$ X $$**
$$
X = A^{-1}B = \frac{1}{-105} \begin{bmatrix}
-200 & -100 \
-0.25 & 0.4
\end{bmatrix}
\begin{bmatrix}
255 \
177
\end{bmatrix}
$$

**Calculations:**
- For $$ Y $$:
$$
Y = \frac{(-200 \times 255) + (-100 \times 177)}{-105} = \frac{-51,000 - 17,700}{-105} = \frac{-68,700}{-105} = \frac{68,700}{105} = 654 \frac{2}{7}
$$
- For $$ I $$:
$$
I = \frac{(-0.25 \times 255) + (0.4 \times 177)}{-105} = \frac{-63.75 + 70.8}{-105} = \frac{7.05}{-105} = -\frac{47}{700} \approx -0.067
$$

**Final Answer for Part (b):**
$$
\begin{cases}
\text{Equilibrium Level of Income } Y = 654 \frac{2}{7} \
\text{Equilibrium Rate of Interest } I = -\frac{47}{700} \approx -0.067
\end{cases}
$$

---

**Summary:**

- **5(a):** $$ x + y = -\frac{31}{11} $$
- **5(b):** 
  - **Equilibrium Income (Y):** $$ 654 \frac{2}{7} $$
  - **Equilibrium Interest Rate (I):** $$ -\frac{47}{700} $$ or approximately $$-0.067$$
**5(a):** $$ x + y = -\frac{31}{11} $$ **5(b):** - **Equilibrium Income (Y):** $$ 654 \frac{2}{7} $$ - **Equilibrium Interest Rate (I):** $$ -\frac{47}{700} $$ or approximately $$-0.067$$

5(a) (a) Using Cramer’s rule to determine , if
(b) Given IS equation and the equation . Use
matrix form to find the equilibrium level of income and rate of interest.

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5(a) (a) Using Cramer’s rule to determine , if (b) Given IS equation and the equation . Use matrix form to find the equilibrium level of income and rate of interest.