Lesson 5: More Linear Equations in Disguise

Property

A system of equations involving square root terms can be solved using the elimination method. Treat terms like $\sqrt{x}$ and $\sqrt{y}$ as variables and manipulate the equations to eliminate one of them, just as you would with a linear system.

Examples

• Given the system: $\sqrt{x} + 2\sqrt{y} = 7$ and $-\sqrt{x} + 3\sqrt{y} = 8$. Adding the two equations yields $5\sqrt{y} = 15$, so $\sqrt{y} = 3$, which means $y = 9$. Substituting back gives $\sqrt{x} + 2(3) = 7$, so $\sqrt{x} = 1$ and $x=1$. The solution is $(1, 9)$.
• Given the system: $3\sqrt{a} + \sqrt{b} = 9$ and $2\sqrt{a} - 2\sqrt{b} = 2$. Multiply the first equation by $2$ to get $6\sqrt{a} + 2\sqrt{b} = 18$. Adding this to the second equation gives $8\sqrt{a} = 20$, so $\sqrt{a} = \frac{5}{2}$ and $a = \frac{25}{4}$. Substituting back gives $3(\frac{5}{2}) + \sqrt{b} = 9$, so $\sqrt{b} = \frac{3}{2}$ and $b = \frac{9}{4}$.

Explanation

This method treats the entire square root expression (e.g., $\sqrt{x}$) as a single variable. By applying the elimination method directly, you can solve for the value of the square root term. This approach bypasses the need for formal variable substitution, offering a more direct path to the solution. After finding the value of the square root term, remember to square it to find the final value of the variable.

Property

When systems contain multiple complex expressions (radicals, reciprocals, or combinations), use sequential substitution: let $u = \sqrt{x}$, $v = \sqrt{y}$, and $w = \frac{1}{x}$ to transform the system into linear form, then solve systematically by back-substitution.

Examples

Property

When solving systems of equations where variables appear in the denominator, one approach is to multiply each equation by the least common denominator (LCD) to eliminate the fractions. This transforms the rational equations into polynomial equations, which may or may not be linear. For example, for an equation $\frac{a}{x} + \frac{b}{y} = c$, multiplying by the LCD $xy$ results in $ay + bx = cxy$.

Examples

• Given the system:

Multiplying both equations by the LCD, $xy$, results in a non-linear system:

• Given the system:

Multiplying the first equation by $y$ gives:

This is now a standard linear system.

Explanation

This method involves clearing the fractions by multiplying each equation by its least common denominator (LCD). While this can simplify equations by removing fractions, it is important to be cautious. This technique can transform a "disguised" linear system into a more complex non-linear system, particularly when the LCD contains multiple variables. This approach is often less efficient than using substitution (e.g., $u = \frac{1}{x}, v = \frac{1}{y}$), but it can be effective in specific cases where it leads to a simpler system.

Property

When a system contains reciprocal expressions like $\frac{1}{x}$ or $\frac{1}{y}$, substitute new variables for these reciprocals to create a linear system. Let $u = \frac{1}{x}$ and $v = \frac{1}{y}$, then solve for $u$ and $v$, and finally find $x = \frac{1}{u}$ and $y = \frac{1}{v}$.

Examples