Question

If \(a / b=5\) and \(a b=5\), what is \(a+b\) ? (A) 6 (B) 5 (C) 4 (D) \(-6\) (E) 6 or \(-6\)

Short answer

6 or -6

Step-by-step solution

Express one variable in terms of the other

From the equation \( \frac{a}{b} = 5 \), express \( a \) in terms of \( b \). We get: $$a = 5b$$.

Substitute into the second equation

Substitute \( a = 5b \) into the equation \( ab = 5 \). We get: $$5b \cdot b = 5$$

Solve for \( b \)

Now we have the equation: $$5b^2 = 5$$. Dividing both sides by 5, we get: $$b^2 = 1$$. So, $$b = 1$$ or $$b = -1$$.

Find corresponding \( a \) values

Using the values obtained for \( b \), find the corresponding \( a \) values. For \( b = 1 \), $$a = 5 \cdot 1 = 5$$. For \( b = -1 \), $$a = 5 \cdot (-1) = -5$$.

Calculate \( a + b \)

Find the values of \( a + b \) for the pairs \( (a, b) = (5, 1) \) and \( (a, b) = (-5, -1) \). They are: $$5 + 1 = 6$$ and $$-5 + (-1) = -6$$.

Key concepts

Equations

In algebra, an equation is a mathematical statement that asserts the equality of two expressions. Consider it a balance scale where both sides should be equal. For example, in the exercise, we have two equations: \(\frac{a}{b} = 5\) and \(ab = 5\).
Understanding how to manipulate these equations is crucial. The first step often involves expressing variables differently to simplify the equation.
Here, \(\frac{a}{b}=5\) is an equation that tells us that the ratio of \(a\) to \(b\) is 5. Converting this to a more workable form, we get \(a = 5b\).
In the second equation, \(ab=5\), substituting \(5b\) for \(a\) allows us to break it down in a more streamlined manner.

Solving for Variables

Solving for variables means finding the value of unknowns in an equation. Starting from the equation \(a = 5b\), we substitute it into \(ab = 5\), resulting in \(5b \cdot b = 5\). This gives us a quadratic equation: \5b^2 = 5\.
Divide both sides by 5 to get \(b^2 = 1\). Taking the square root of both sides gives two possible values for \(b\), \(b = 1\) and \(b = -1\).
Once we have these values, we can substitute them back into the original relationship \(a = 5b\) to find the values of \(a\): \(b = 1\) results in \(a = 5 \times 1 = 5\), and \(b = -1\) gives \(a = 5 \times -1 = -5\).
Finally, by substituting our results back, we verify that our values satisfy both original equations.

Substitution

Substitution is a core technique in algebra for solving systems of equations. It involves replacing one variable with an expression based on another variable.
In our exercise, we first express \(a\) in terms of \(b\) as \(a = 5b\). We then replace \(a\) in the second equation \(ab=5\), giving \(5b \times b = 5\), which simplifies our problem significantly.
This strategy simplifies more complicated problems involving multiple steps.
After substituting and solving the equation for one variable, we substitute back again to find the other variable. This confirms that both values satisfy the original equations.