Question

If \(|2 x-5|=x+13\) and \(|4-3 y|=2,\) what is the largest possible value of \(\frac{y}{x} ?\)

Short answer

The largest possible value of \(\frac{y}{x}\) is \(\frac{1}{9}\).

Step-by-step solution

Solve the first absolute value equation

Given \[|2x - 5| = x + 13\], we need to consider two cases: Case 1: \[2x - 5 = x + 13\] Case 2: \[2x - 5 = - (x + 13)\]

Solve Case 1

For the equation \[2x - 5 = x + 13\]: Subtract \[x\] from both sides: \[2x - 5 - x = x + 13 - x\] Simplify: \[x - 5 = 13\] Add 5 to both sides: \[x = 18\]

Solve Case 2

For the equation \[2x - 5 = - (x + 13)\]: Distribute the negative sign: \[2x - 5 = -x - 13\] Add \[x\] to both sides: \[2x + x - 5 = -x + x - 13\] Simplify: \[3x - 5 = -13\] Add 5 to both sides: \[3x = -8\] Divide by 3: \[x = -\frac{8}{3}\]

Solve the second absolute value equation

Given \[|4 - 3y| = 2\], we need to consider two cases: Case 1: \[4 - 3y = 2\] Case 2: \[4 - 3y = -2\]

Solve second equation Case 1

For the equation \[4 - 3y = 2\]: Subtract 4 from both sides: \[4 - 3y - 4 = 2 - 4\] Simplify: \[-3y = -2\] Divide by -3: \[y = \frac{2}{3}\]

Solve second equation Case 2

For the equation \[4 - 3y = -2\]: Subtract 4 from both sides: \[4 - 3y - 4 = -2 - 4\] Simplify: \[-3y = -6\] Divide by -3: \[y = 2\]

Determine the ratio \(\frac{y}{x}\) for both solutions

We have found \[x = 18\] and \[x = -\frac{8}{3}\], as well as \[y = \frac{2}{3}\] and \[y = 2\]. For \[x = 18\] and \[y = \frac{2}{3}\], the ratio is \(\frac{\frac{2}{3}}{18} = \frac{1}{27}\). For \[x = 18\] and \[y = 2\], the ratio is \(\frac{2}{18} = \frac{1}{9}\). For \[x = -\frac{8}{3}\] and \[y = \frac{2}{3}\], the ratio is \(\frac{\frac{2}{3}}{-\frac{8}{3}} = -\frac{1}{4}\). For \[x = -\frac{8}{3}\] and \[y = 2\], the ratio is \(\frac{2}{-\frac{8}{3}} = -\frac{3}{4}\).

Determine the largest possible value of \(\frac{y}{x}\)

Based on the calculations, the largest value is \(\frac{1}{9}\)

Key concepts

Absolute Value Equations

Absolute value equations involve solving for a variable within an absolute value expression. The absolute value of a number is its distance from zero on the number line, without regard to direction. Therefore, any absolute value equation such as \(|2x - 5| = x + 13\) is solved by considering two possible cases: the expression inside the absolute value being positive or being negative.

For instance, solving \(|2x - 5| = x + 13\) involves two scenarios:
1. The expression \(2x - 5\) is equal to \(x + 13\). This is considered as: \(2x - 5 = x + 13\).
2. The expression \(2x - 5\) is the negative of \(x + 13\). This is considered as: \(2x - 5 = - (x + 13)\).

Each scenario is then solved like a normal linear equation.
When dealing with absolute value equations, always break them into these two simpler cases as a first step.

Solving Equations

Solving equations is the process of finding the value of a variable that makes the equation true. For linear equations, the goal is to isolate the variable on one side of the equation.
Let’s use one of the equations we discussed earlier as an example: \(2x - 5 = - (x + 13)\). The steps for solving this are:
1. Distribute any negative signs or constants: This means changing \(- (x + 13)\) to \(-x - 13\).
2. Combine like terms: For example, adding \x\ to both sides here leads to \(2x + x - 5 = -13\) which simplifies to \(3x - 5 = -13\).
3. Isolate the variable: Add 5 to both sides: \(3x = -8\). Finally, divide by 3: \(x = -\frac{8}{3}\).
Different methods are required for different types of equations, but the principle of isolating the variable remains constant.

Ratios

Ratios compare two quantities, showing the relative size of one compared to the other. A common task in algebra is to find the ratio between variables.
In the given exercise, we were asked to find the ratio \(\frac{y}{x}\) once we solved for both variables.

Given \(x = 18\) and \(y = 2\), for instance, the ratio \(\frac{y}{x} = \frac{2}{18} = \frac{1}{9}\). Similarly, each calculated \(x\) and \(y\) yields a different ratio, which we then compare to determine the largest possible value.

Ratios are simplified to their smallest form by dividing the numerator and the denominator by their greatest common divisor. Thus, \(\frac{2}{18}\) simplifies to \(\frac{1}{9}\).
Understanding how to manipulate ratios is crucial for comparing, scaling, or finding relationships between quantities.