Given: \( a^{2}-5 a b+4 b^{2}=0 \). (1) Solve for \( a \) in terms of \( b \). (2) Hence or otherwise determine the values(s) of the ratio \( \frac{a}{b} \). (3) If the sum of \( a \) and \( b \) is 2 , determine the values of \( a \) and \( b \).

Solve the equation by following steps: - step0: Solve for \(a\): \(a^{2}-5ab+4b^{2}=0\) - step1: Rewrite the expression: \(a^{2}-5ba+4b^{2}=0\) - step2: Factor the expression: \(\left(a-b\right)\left(a-4b\right)=0\) - step3: Separate into possible cases: \(\begin{align}&a-b=0\\&a-4b=0\end{align}\) - step4: Solve the equation: \(\begin{align}&a=b\\&a=4b\end{align}\) Solve the equation \( a=b \). Solve the equation by following steps: - step0: Solve for \(b\): \(a=b\) - step1: Swap the sides: \(b=a\) Solve the equation \( a=4b \). Solve the equation by following steps: - step0: Solve for \(b\): \(a=4b\) - step1: Swap the sides: \(4b=a\) - step2: Divide both sides: \(\frac{4b}{4}=\frac{a}{4}\) - step3: Divide the numbers: \(b=\frac{a}{4}\) Solve the system of equations \( a+b=2;a=b \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}a+b=2\\a=b\end{array}\right.\) - step1: Substitute the value of \(a:\) \(b+b=2\) - step2: Simplify: \(2b=2\) - step3: Divide both sides: \(\frac{2b}{2}=\frac{2}{2}\) - step4: Divide the numbers: \(b=1\) - step5: Substitute the value of \(b:\) \(a=1\) - step6: Calculate: \(\left\{ \begin{array}{l}a=1\\b=1\end{array}\right.\) - step7: Check the solution: \(\left\{ \begin{array}{l}a=1\\b=1\end{array}\right.\) - step8: Rewrite: \(\left(a,b\right) = \left(1,1\right)\) Solve the system of equations \( a+b=2;a=4b \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}a+b=2\\a=4b\end{array}\right.\) - step1: Substitute the value of \(a:\) \(4b+b=2\) - step2: Simplify: \(5b=2\) - step3: Divide both sides: \(\frac{5b}{5}=\frac{2}{5}\) - step4: Divide the numbers: \(b=\frac{2}{5}\) - step5: Substitute the value of \(b:\) \(a=4\times \frac{2}{5}\) - step6: Calculate: \(a=\frac{8}{5}\) - step7: Calculate: \(\left\{ \begin{array}{l}a=\frac{8}{5}\\b=\frac{2}{5}\end{array}\right.\) - step8: Check the solution: \(\left\{ \begin{array}{l}a=\frac{8}{5}\\b=\frac{2}{5}\end{array}\right.\) - step9: Rewrite: \(\left(a,b\right) = \left(\frac{8}{5},\frac{2}{5}\right)\) Let's solve the problem step by step. ### Step 1: Solve for \( a \) in terms of \( b \) We start with the equation: \[ a^{2} - 5ab + 4b^{2} = 0 \] This is a quadratic equation in terms of \( a \). Using the quadratic formula \( a = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \), where \( A = 1 \), \( B = -5b \), and \( C = 4b^2 \): 1. Calculate the discriminant: \[ B^2 - 4AC = (-5b)^2 - 4(1)(4b^2) = 25b^2 - 16b^2 = 9b^2 \] 2. Substitute into the quadratic formula: \[ a = \frac{5b \pm 3b}{2} \] This gives us two solutions: \[ a = \frac{8b}{2} = 4b \quad \text{and} \quad a = \frac{2b}{2} = b \] Thus, we have: \[ a = b \quad \text{or} \quad a = 4b \] ### Step 2: Determine the values of the ratio \( \frac{a}{b} \) From the solutions: 1. If \( a = b \), then: \[ \frac{a}{b} = 1 \] 2. If \( a = 4b \), then: \[ \frac{a}{b} = 4 \] Thus, the possible values of the ratio \( \frac{a}{b} \) are \( 1 \) and \( 4 \). ### Step 3: If the sum of \( a \) and \( b \) is 2, determine the values of \( a \) and \( b \) We have two cases to consider based on the values of \( a \) in terms of \( b \). #### Case 1: \( a = b \) Substituting into the sum: \[ a + b = 2 \implies b + b = 2 \implies 2b = 2 \implies b = 1 \] Thus, \( a = 1 \). So, one solution is: \[ (a, b) = (1, 1) \] #### Case 2: \( a = 4b \) Substituting into the sum: \[ a + b = 2 \implies 4b + b = 2 \implies 5b = 2 \implies b = \frac{2}{5} \] Thus, \( a = 4b = 4 \cdot \frac{2}{5} = \frac{8}{5} \). So, another solution is: \[ (a, b) = \left(\frac{8}{5}, \frac{2}{5}\right) \quad \text{or} \quad (1.6, 0.4) \] ### Final Answers 1. \( a = b \) or \( a = 4b \) 2. The values of the ratio \( \frac{a}{b} \) are \( 1 \) and \( 4 \). 3. The values of \( (a, b) \) are: - \( (1, 1) \) - \( \left(\frac{8}{5}, \frac{2}{5}\right) \) or \( (1.6, 0.4) \)