Solving the Simultaneous Equations a*b 40 and a - b 5: A Step-by-Step Guide

Introduction

Algebra is a fundamental branch of mathematics that often deals with solving equations. One common type of algebraic problem involves simultaneous equations, where you are given a set of equations with multiple variables and need to find the values of these variables. This article will guide you through solving the simultaneous equations a*b 40 and a - b 5, providing a clear and detailed explanation.

The Problem

The problem at hand is given by two equations:

The first equation is a*b 40, which means the product of a and b equals 40. The second equation is a - b 5, indicating that the difference between a and b is 5.

Solving the Equations

Let's solve this step-by-step:

Step 1: Express a in terms of b or vice versa

From the second equation, we can express a in terms of b:

a - b 5

Adding b to both sides of the equation gives:

a b 5

Step 2: Substitute the expression for a into the first equation

Now that we have a in terms of b, we can substitute it into the first equation:

(b 5) * b 40

Multiplying out the left side of the equation:

b^2 5b 40

Step 3: Rearrange into a standard quadratic equation

Subtract 40 from both sides to bring the equation to the standard quadratic form:

b^2 5b - 40 0

Step 4: Solve the quadratic equation

We can solve this quadratic equation using the quadratic formula, which states that for an equation of the form ax^2 bx c 0, the solutions are given by:

x [-b ± sqrt(b^2 - 4ac)] / (2a)

For our equation, a 1, b 5, and c -40. Substituting these values into the quadratic formula:

b [-5 ± sqrt(5^2 - 4(1)(-40))] / (2(1))

b [-5 ± sqrt(25 160)] / 2

b [-5 ± sqrt(185)] / 2

Since we only need the positive value for b (because a and b are real numbers and their product is positive), we take the positive root:

b [-5 sqrt(185)] / 2

This simplifies to approximately:

b ≈ 4.18

Step 5: Find the value of a

Now that we have the value of b, we can use the first equation, a b 5, to find a:

a 4.18 5

a ≈ 9.18

Verification and Summary

We can verify that these values satisfy both the original equations:

(9.18) * (4.18) ≈ 40 9.18 - 4.18 5

Therefore, the solution to the system of equations a*b 40 and a - b 5 is a ≈ 9.18 and b ≈ 4.18.

Additional Insights

Alternative Method

Another approach is to add the two equations directly:

a*b 40

a - b 5

From a - b 5, express a as: a b 5

Substitute a b 5 into a*b 40:

(b 5) * b 40

b^2 5b 40

b^2 5b - 40 0

Using the quadratic formula, solve for b ≈ 4.18

Then, a b 5 ≈ 9.18

Conclusion

In conclusion, solving the simultaneous equations a*b 40 and a - b 5 yields the solution a ≈ 9.18 and b ≈ 4.18. This method involves expressing one variable in terms of the other, substituting into the second equation, and then solving the resulting quadratic equation. Through this step-by-step process, we can confidently find the values of a and b that satisfy both equations.