Problem 20 - Precalculus

Question

Solve the following system $$ \left\\{\begin{aligned} x^{2}+\sqrt{y}+\log _{2}(z) &=6 \\ 3 x^{2}-2 \sqrt{y}+2 \log _{2}(z) &=5 \\ -5 x^{2}+3 \sqrt{y}+4 \log _{2}(z) &=13 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution requires multiple iterative trials to find consistent variables; testing typically around particular substitution constraints reveals connected roots.

Step 1: Simplify the equations

The given system of equations is: \(\begin{align*} 1.\ & \ x^2 + \sqrt{y} + \log_2(z) = 6 \2.\ & \ 3x^2 - 2\sqrt{y} + 2\log_2(z) = 5 \3.\ & \ -5x^2 + 3\sqrt{y} + 4\log_2(z) = 13 \end{align*}\)First, we can express \(\sqrt{y}\) and \(\log_2(z)\) in terms of \(x^2\) from equations 1 and 2, then substitute into equation 3. These manipulations are necessary since we have three variables and need to find a common expression to help simplify the equations.

Step 2: Express \(\sqrt{y}\) and \(\log_2(z)\)

From equation 1, express \(\sqrt{y}\) and \(\log_2(z)\) using \(x^2\): \[ \sqrt{y} + \log_2(z) = 6 - x^2 \]From equation 2:\[ -2\sqrt{y} + 2\log_2(z) = 5 - 3x^2 \]We can express \(\sqrt{y}\) and \(\log_2(z)\) in terms of each other, or solve these two for one of these variables and substitute into equation 3. To facilitate this step, solve equation 2 for \(\sqrt{y}\):\[ \sqrt{y} = \log_2(z) - \frac{5}{2} + \frac{3x^2}{2} \]

Step 3: Substitute into Third Equation

Substitute expressions for \(\sqrt{y}\) and \(\log_2(z)\) into equation 3:\[-5x^2 + 3\left(\log_2(z) - \frac{5}{2} + \frac{3x^2}{2}\right) + 4\log_2(z) = 13\]Simplify:\[-5x^2 + 3\log_2(z) - \frac{15}{2} + \frac{9x^2}{2} + 4\log_2(z) = 13\]Combine like terms:\[-5x^2 + \frac{9x^2}{2} + 7\log_2(z) - \frac{15}{2} = 13\]

Step 4: Solve resulting equation

To solve:\[\frac{-10x^2 + 9x^2}{2} + 7\log_2(z) = 13 + \frac{15}{2}\]\[\frac{-x^2}{2} + 7\log_2(z) = \frac{41}{2}\]Multiply through by 2:\[-x^2 + 14\log_2(z) = 41\]Express \(\log_2(z)\) in terms of \(x^2\):\[\log_2(z) = \frac{x^2 + 41}{14}\]

Step 5: Find consistent solutions

Substitute \(\log_2(z)\) back into expressions from Step 2:\[\sqrt{y} = \frac{x^2 + 41}{14} - \frac{5}{2} + \frac{3x^2}{2}\]\[\sqrt{y} = \frac{x^2 + 41 - 35 + 21x^2}{14}\]\[\sqrt{y} = \frac{22x^2 + 6}{14} \]Simplify:\[\sqrt{y} = \frac{11x^2 + 3}{7}\]Using substitution values for small integers, test for whole number results that can fit these simplifications. Usually trying simple cases can help -- values \(x=1\) for example, need checking. If these fit and satisfy the equations, confirm correctness with each initial equation.

Step 6: Verification and Possible Solutions

Using substitutions, we verify with trial and error. For instance testing with \(x = 1\) results in checking what constraints are satisfied and modifying accordingly on each variable manipulative and consistency; notes of roots must be checked against significant constraints on all logical variables.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is an essential skill in solving systems of equations. It involves rearranging and simplifying expressions to find the values of unknown variables. In this problem, we used algebraic manipulation to express the terms \(\sqrt{y}\) and \(\log_2(z)\) in terms of \(x^2\). This is done by analyzing the relationships in the first two equations, allowing us to substitute into the third equation.

The objective of these manipulations is to combine and eliminate variables until a single equation with one unknown remains. This can be achieved by isolating one variable in terms of others, then substituting back into the equations. The process often involves combining like terms and multiplying or dividing through by constants to simplify the resulting expressions.

As you work through these equations, always check for opportunities to factor out common elements. This can particularly simplify expressions and reveal solutions faster. Mastery in algebraic manipulation builds the foundation for solving more complex systems.

Logarithmic Equations

Logarithmic equations in algebra involve the logarithm of a variable, often essential in exponential growth or decay contexts. In the exercise, the term \(\log_2(z)\) appears repeatedly. The core principle of logarithmic equations is understanding how to manipulate them, mainly the laws of logs, such as the product, quotient, and power rules.

One useful property is that logarithms convert multiplicative relationships into additive ones, which can simplify equations significantly. For instance, considering

• \(\log_b(mn) = \log_b(m) + \log_b(n)\)
• \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\)

can help unravel complex expressions involving logarithmic terms.

When rearranging the equations to express \(\log_2(z)\) as a function of \(x^2\), it's crucial to use these properties consistently. Simplifying logarithms often involves converting the equations into an exponential form or vice versa, allowing one to make the equations linear and therefore easier to solve.

Radical Equations

Radical equations contain terms with roots, most commonly the square roots seen in this problem as \(\sqrt{y}\). Solving these equations often requires isolating the radical on one side and then squaring both sides of the equation to eliminate the square root. However, squaring can introduce extraneous solutions, which are solutions that don't satisfy the original equation. Thus, it's essential to check solutions at the end.

Steps to solving typically include:

• Isolating the radical term if possible, such as in \(\sqrt{y} = f(x)\).
• Squaring both sides to remove the square root, being careful as this can introduce extra solutions.

If the radical is not easily isolated, you may need to first express it with other terms, performing operations that simplify its integration with the overall system, as done in this exercise.

For successful handling of radical equations in systems like this one, practice how to express one variable in terms of others, and always circle back to initial conditions to verify solutions thoroughly.