Answer :
Sure, I can show you a detailed, step-by-step solution to solve the system of equations:
1)
[tex]\[ \begin{cases} 3a + b = 2 \\ 9a^2 - b^2 = 7 \end{cases} \][/tex]
Step 1: Express [tex]\( b \)[/tex] in terms of [tex]\( a \)[/tex] using the first equation.
[tex]\[ 3a + b = 2 \quad \Rightarrow \quad b = 2 - 3a \][/tex]
Step 2: Substitute [tex]\( b = 2 - 3a \)[/tex] into the second equation.
[tex]\[ 9a^2 - (2 - 3a)^2 = 7 \][/tex]
Step 3: Simplify the equation.
[tex]\[ 9a^2 - (4 - 12a + 9a^2) = 7 \quad \Rightarrow \quad 9a^2 - 4 + 12a - 9a^2 = 7 \quad \Rightarrow \quad 12a - 4 = 7 \][/tex]
Step 4: Solve for [tex]\( a \)[/tex].
[tex]\[ 12a - 4 = 7 \quad \Rightarrow \quad 12a = 11 \quad \Rightarrow \quad a = \frac{11}{12} \][/tex]
Step 5: Substitute [tex]\( a = \frac{11}{12} \)[/tex] back into [tex]\( b = 2 - 3a \)[/tex] to find [tex]\( b \)[/tex].
[tex]\[ b = 2 - 3\left(\frac{11}{12}\right) \quad \Rightarrow \quad b = 2 - \frac{33}{12} \quad \Rightarrow \quad b = 2 - \frac{11}{4} \quad \Rightarrow \quad b = \frac{8}{4} - \frac{11}{4} \quad \Rightarrow \quad b = -\frac{3}{4} \][/tex]
So, the solution to the system of equations is:
[tex]\[ \left( a, b \right) = \left( \frac{11}{12}, -\frac{3}{4} \right) \][/tex]
1)
[tex]\[ \begin{cases} 3a + b = 2 \\ 9a^2 - b^2 = 7 \end{cases} \][/tex]
Step 1: Express [tex]\( b \)[/tex] in terms of [tex]\( a \)[/tex] using the first equation.
[tex]\[ 3a + b = 2 \quad \Rightarrow \quad b = 2 - 3a \][/tex]
Step 2: Substitute [tex]\( b = 2 - 3a \)[/tex] into the second equation.
[tex]\[ 9a^2 - (2 - 3a)^2 = 7 \][/tex]
Step 3: Simplify the equation.
[tex]\[ 9a^2 - (4 - 12a + 9a^2) = 7 \quad \Rightarrow \quad 9a^2 - 4 + 12a - 9a^2 = 7 \quad \Rightarrow \quad 12a - 4 = 7 \][/tex]
Step 4: Solve for [tex]\( a \)[/tex].
[tex]\[ 12a - 4 = 7 \quad \Rightarrow \quad 12a = 11 \quad \Rightarrow \quad a = \frac{11}{12} \][/tex]
Step 5: Substitute [tex]\( a = \frac{11}{12} \)[/tex] back into [tex]\( b = 2 - 3a \)[/tex] to find [tex]\( b \)[/tex].
[tex]\[ b = 2 - 3\left(\frac{11}{12}\right) \quad \Rightarrow \quad b = 2 - \frac{33}{12} \quad \Rightarrow \quad b = 2 - \frac{11}{4} \quad \Rightarrow \quad b = \frac{8}{4} - \frac{11}{4} \quad \Rightarrow \quad b = -\frac{3}{4} \][/tex]
So, the solution to the system of equations is:
[tex]\[ \left( a, b \right) = \left( \frac{11}{12}, -\frac{3}{4} \right) \][/tex]
