Answer :
Let's solve for the variables [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(e\)[/tex], and [tex]\(f\)[/tex] using the given equations. We will proceed step-by-step through each equation and solve the system of equations.
The given equations are:
1. [tex]\(a + b = 10\)[/tex]
2. [tex]\(e - f = 5\)[/tex]
3. [tex]\(a \times e = 20\)[/tex]
4. [tex]\(b + f = 15\)[/tex]
We need to find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(e\)[/tex], and [tex]\(f\)[/tex].
### Step-by-Step Solution:
Step 1: Solve for [tex]\(b\)[/tex] in terms of [tex]\(a\)[/tex]
From the first equation:
[tex]\[ a + b = 10 \][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[ b = 10 - a \][/tex]
Step 2: Solve for [tex]\(f\)[/tex] in terms of [tex]\(b\)[/tex]
From the fourth equation:
[tex]\[ b + f = 15 \][/tex]
Substitute [tex]\( b = 10 - a \)[/tex]:
[tex]\[ (10 - a) + f = 15 \][/tex]
Solving for [tex]\(f\)[/tex]:
[tex]\[ f = 15 - (10 - a) \][/tex]
[tex]\[ f = 5 + a \][/tex]
Step 3: Solve for [tex]\(e\)[/tex] in terms of [tex]\(a\)[/tex]
From the third equation:
[tex]\[ a \times e = 20 \][/tex]
Solving for [tex]\(e\)[/tex]:
[tex]\[ e = \frac{20}{a} \][/tex]
Step 4: Solve for [tex]\(e\)[/tex] using the second equation
From the second equation:
[tex]\[ e - f = 5 \][/tex]
Substitute [tex]\( f = 5 + a \)[/tex] and [tex]\( e = \frac{20}{a} \)[/tex]:
[tex]\[ \frac{20}{a} - (5 + a) = 5 \][/tex]
Simplifying the equation:
[tex]\[ \frac{20}{a} - 5 - a = 5 \][/tex]
[tex]\[ \frac{20}{a} - a = 10 \][/tex]
Step 5: Solve the simplified equation for [tex]\(a\)[/tex]
[tex]\[ \frac{20}{a} - a = 10 \][/tex]
Multiply both sides by [tex]\(a\)[/tex] to eliminate the fraction:
[tex]\[ 20 - a^2 = 10a \][/tex]
[tex]\[ 20 = 10a + a^2 \][/tex]
Rearrange the equation:
[tex]\[ a^2 + 10a - 20 = 0 \][/tex]
Step 6: Solve the quadratic equation
To solve the quadratic equation [tex]\(a^2 + 10a - 20 = 0\)[/tex], we'll use the quadratic formula [tex]\(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex]:
In this case:
[tex]\[ a = 1, \quad b = 10, \quad c = -20 \][/tex]
[tex]\[ a = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot (-20)}}{2 \cdot 1} \][/tex]
[tex]\[ a = \frac{-10 \pm \sqrt{100 + 80}}{2} \][/tex]
[tex]\[ a = \frac{-10 \pm \sqrt{180}}{2} \][/tex]
[tex]\[ a = \frac{-10 \pm 6\sqrt{5}}{2} \][/tex]
[tex]\[ a = -5 \pm 3\sqrt{5} \][/tex]
Thus, we have two possible values for [tex]\( a \)[/tex]:
[tex]\[ a = -5 + 3\sqrt{5} \quad \text{or} \quad a = -5 - 3\sqrt{5} \][/tex]
Step 7: Calculate corresponding values of [tex]\(b\)[/tex], [tex]\(e\)[/tex], and [tex]\(f\)[/tex]
For [tex]\( a = -5 + 3\sqrt{5} \)[/tex]:
- [tex]\( b = 10 - a = 10 - (-5 + 3\sqrt{5}) = 15 - 3\sqrt{5} \)[/tex]
- [tex]\( e = \frac{20}{a} = \frac{20}{-5 + 3\sqrt{5}} \)[/tex]
- [tex]\( f = 5 + a = 5 + (-5 + 3\sqrt{5}) = 3\sqrt{5} \)[/tex]
For [tex]\( a = -5 - 3\sqrt{5} \)[/tex]:
- [tex]\( b = 10 - a = 10 - (-5 - 3\sqrt{5}) = 15 + 3\sqrt{5} \)[/tex]
- [tex]\( e = \frac{20}{a} = \frac{20}{-5 - 3\sqrt{5}} \)[/tex]
- [tex]\( f = 5 + a = 5 + (-5 - 3\sqrt{5}) = -3\sqrt{5} \)[/tex]
Since [tex]\( e = \frac{20}{a} \rightarrow \frac{20}{-5 + 3\sqrt{5}} \)[/tex] and [tex]\( \frac{20}{-5 - 3\sqrt{5}} \)[/tex] results in more complex expressions, I'll leave these calculations for you to explore further based on these.
Therefore, the solutions are:
1. [tex]\( a = -5 + 3\sqrt{5} \)[/tex], [tex]\( b = 15 - 3\sqrt{5} \)[/tex], [tex]\( e = \frac{20}{-5 + 3\sqrt{5}} \)[/tex], [tex]\( f = 3\sqrt{5} \)[/tex]
2. [tex]\( a = -5 - 3\sqrt{5} \)[/tex], [tex]\( b = 15 + 3\sqrt{5} \)[/tex], [tex]\( e = \frac{20}{-5 - 3\sqrt{5}} \)[/tex], [tex]\( f = -3\sqrt{5} \)[/tex]
The given equations are:
1. [tex]\(a + b = 10\)[/tex]
2. [tex]\(e - f = 5\)[/tex]
3. [tex]\(a \times e = 20\)[/tex]
4. [tex]\(b + f = 15\)[/tex]
We need to find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(e\)[/tex], and [tex]\(f\)[/tex].
### Step-by-Step Solution:
Step 1: Solve for [tex]\(b\)[/tex] in terms of [tex]\(a\)[/tex]
From the first equation:
[tex]\[ a + b = 10 \][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[ b = 10 - a \][/tex]
Step 2: Solve for [tex]\(f\)[/tex] in terms of [tex]\(b\)[/tex]
From the fourth equation:
[tex]\[ b + f = 15 \][/tex]
Substitute [tex]\( b = 10 - a \)[/tex]:
[tex]\[ (10 - a) + f = 15 \][/tex]
Solving for [tex]\(f\)[/tex]:
[tex]\[ f = 15 - (10 - a) \][/tex]
[tex]\[ f = 5 + a \][/tex]
Step 3: Solve for [tex]\(e\)[/tex] in terms of [tex]\(a\)[/tex]
From the third equation:
[tex]\[ a \times e = 20 \][/tex]
Solving for [tex]\(e\)[/tex]:
[tex]\[ e = \frac{20}{a} \][/tex]
Step 4: Solve for [tex]\(e\)[/tex] using the second equation
From the second equation:
[tex]\[ e - f = 5 \][/tex]
Substitute [tex]\( f = 5 + a \)[/tex] and [tex]\( e = \frac{20}{a} \)[/tex]:
[tex]\[ \frac{20}{a} - (5 + a) = 5 \][/tex]
Simplifying the equation:
[tex]\[ \frac{20}{a} - 5 - a = 5 \][/tex]
[tex]\[ \frac{20}{a} - a = 10 \][/tex]
Step 5: Solve the simplified equation for [tex]\(a\)[/tex]
[tex]\[ \frac{20}{a} - a = 10 \][/tex]
Multiply both sides by [tex]\(a\)[/tex] to eliminate the fraction:
[tex]\[ 20 - a^2 = 10a \][/tex]
[tex]\[ 20 = 10a + a^2 \][/tex]
Rearrange the equation:
[tex]\[ a^2 + 10a - 20 = 0 \][/tex]
Step 6: Solve the quadratic equation
To solve the quadratic equation [tex]\(a^2 + 10a - 20 = 0\)[/tex], we'll use the quadratic formula [tex]\(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex]:
In this case:
[tex]\[ a = 1, \quad b = 10, \quad c = -20 \][/tex]
[tex]\[ a = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot (-20)}}{2 \cdot 1} \][/tex]
[tex]\[ a = \frac{-10 \pm \sqrt{100 + 80}}{2} \][/tex]
[tex]\[ a = \frac{-10 \pm \sqrt{180}}{2} \][/tex]
[tex]\[ a = \frac{-10 \pm 6\sqrt{5}}{2} \][/tex]
[tex]\[ a = -5 \pm 3\sqrt{5} \][/tex]
Thus, we have two possible values for [tex]\( a \)[/tex]:
[tex]\[ a = -5 + 3\sqrt{5} \quad \text{or} \quad a = -5 - 3\sqrt{5} \][/tex]
Step 7: Calculate corresponding values of [tex]\(b\)[/tex], [tex]\(e\)[/tex], and [tex]\(f\)[/tex]
For [tex]\( a = -5 + 3\sqrt{5} \)[/tex]:
- [tex]\( b = 10 - a = 10 - (-5 + 3\sqrt{5}) = 15 - 3\sqrt{5} \)[/tex]
- [tex]\( e = \frac{20}{a} = \frac{20}{-5 + 3\sqrt{5}} \)[/tex]
- [tex]\( f = 5 + a = 5 + (-5 + 3\sqrt{5}) = 3\sqrt{5} \)[/tex]
For [tex]\( a = -5 - 3\sqrt{5} \)[/tex]:
- [tex]\( b = 10 - a = 10 - (-5 - 3\sqrt{5}) = 15 + 3\sqrt{5} \)[/tex]
- [tex]\( e = \frac{20}{a} = \frac{20}{-5 - 3\sqrt{5}} \)[/tex]
- [tex]\( f = 5 + a = 5 + (-5 - 3\sqrt{5}) = -3\sqrt{5} \)[/tex]
Since [tex]\( e = \frac{20}{a} \rightarrow \frac{20}{-5 + 3\sqrt{5}} \)[/tex] and [tex]\( \frac{20}{-5 - 3\sqrt{5}} \)[/tex] results in more complex expressions, I'll leave these calculations for you to explore further based on these.
Therefore, the solutions are:
1. [tex]\( a = -5 + 3\sqrt{5} \)[/tex], [tex]\( b = 15 - 3\sqrt{5} \)[/tex], [tex]\( e = \frac{20}{-5 + 3\sqrt{5}} \)[/tex], [tex]\( f = 3\sqrt{5} \)[/tex]
2. [tex]\( a = -5 - 3\sqrt{5} \)[/tex], [tex]\( b = 15 + 3\sqrt{5} \)[/tex], [tex]\( e = \frac{20}{-5 - 3\sqrt{5}} \)[/tex], [tex]\( f = -3\sqrt{5} \)[/tex]
