Methods of solution

Solve the following system of linear equations using the elimination method:

$3x - 2y = 7$

$5x + 4y = 8$

Find the value of the unknowns $x$, $y$, $z$ in the following system of linear equations, which is already in echelon form:

$2x - y + 3z = 14$

$3y - z = 5$

$2z = 8$

Solve the following exponential system of equations by expressing both sides of each equation as powers with the same base:

$2^{x+y} = 32$

$3^{x-y} = 9$

Apply the definition of a logarithm to set up an equivalent linear system and solve the following system of equations:

$\log_2(x + y) = 3$

$x - 2y = 2$

Solve the following system of exponential equations using the changes of variable $u = 3^x$ and $v = 2^y$:

$3^x + 2^{y+1} = 17$

$3^{x-1} - 2^y = 1$

Find the values of $x$ and $y$ in the following system of logarithmic equations, applying logarithmic properties beforehand to simplify the expressions:

$\log(x) + \log(y) = 2$

$\log(x^2) - \log(y) = 1$

The goal is to solve the following system of logarithmic equations over the field of real numbers:

Two students advocate different starting strategies:

• Sara directly applies the power property of logarithms, writes $\log(x^2) = 2\log(x)$, and uses the change of variables $u = \log(x)$, $v = \log(y)$, thereby obtaining only the solution $(x, y) = (10\sqrt{10}, 10)$.

• David argues that applying that property in this specific system reduces the domain of validity and causes real solutions to be lost. He proposes keeping the block $\log(x^2)$ intact and applying the change of variables $u = \log(x^2)$, $v = \log(y)$.

1. a) Determine the domain of definition of the variables $x$ and $y$ in the original system and explain with mathematical rigor why Sara's step is not an equivalence transformation over that entire domain.
2. b) Solve the system following David's approach to find all exact real solutions.
3. c) Conclude how many real solutions the original system has and write the correct general relation between $\log(x^2)$ and $\log|x|$ to avoid this type of error in high school.

Find the solution to the following system of equations involving radical expressions:

$\sqrt{x} + \sqrt{y} = 5$

$x - y = 5$

Consider the following non-linear system with radicals:

1. a) Before working analytically, make a reasoned estimate of the solution:

• Algebraically deduce why $x$ must be strictly greater than $5$ and at the same time less than $49$.

• Knowing from the system that $x + \sqrt{x} = 12$, estimate the exact or approximate value of $x$ by testing simple perfect squares that satisfy this bound. What estimated value corresponds to $y$?

1. b) Solve the system algebraically by substitution or by using the change of variables $u = \sqrt{x}$.
2. c) Check all algebraic solutions obtained in the original system, determine if any of them is extraneous, and assess whether the final result matches your initial estimate.

Cian solves the following system of equations using a change of variables:

To do so, he defines $u = \dfrac{1}{x}$ and $v = \dfrac{1}{y}$, transforming it into a linear system. After applying the elimination method, he flawlessly calculates that $u = 2$ and $v = 1$. However, in the final step, he concludes: "Since $u = 2$ and $v = 1$, the solution to the original system is $x = 2$ and $y = 1$".

1. a) Identify and explain in detail the conceptual error Cian made in his final deduction.
2. b) Correctly undo the change of variables and find the real solution $(x, y)$ of the system.
3. c) Numerically check the obtained solution by substituting it into both equations of the original system.

Solve the following system of equations containing algebraic fractions, clearly stating the initial restrictions on the unknowns:

$\frac{2}{x} + \frac{3}{y} = 2$

$\frac{4}{x} - \frac{1}{y} = \frac{1}{6}$

Apply Gaussian elimination to transform and solve the following system of three linear equations in three unknowns:

$x + y + z = 6$

$x - y + 2z = 5$

$x + y - z = 0$

Solve the following system of linear equations using Gaussian elimination and state whether it is a consistent independent system:

$2x - y + 3z = 1$

$x + 2y - z = 4$

$4x - y + z = 2$

Discuss the following system of linear equations depending on the parameter $k$:

$x + 2y = 5$

$3x + ky = 15$

Find the value of $k$ for which the system has infinitely many solutions and express the general solution in terms of a real parameter $\lambda$.

Solve the following system of three equations in three unknowns using Gaussian elimination. Verify that it is a consistent dependent system and express its solution set in terms of a parameter $\lambda$:

$x - 2y + z = 3$

$2x + y - 3z = 1$

$4x - 3y - z = 7$

Analyse the following system of linear equations and prove, using Gaussian elimination steps, that it is an inconsistent system:

$x + y + z = 4$

$2x - y + 3z = 5$

$3x + 4z = 10$

Solve the following non-linear system of equations using the substitution method:

$x - y = 2$

$x^2 + y^2 = 10$

Find all solutions to the following non-linear system of equations, in which both equations are quadratic:

$x^2 + y^2 - 4x = 5$

$x^2 - y^2 - 4x = -3$

Solve the following symmetric non-linear system of equations. To make it easier to solve, use the change of variables $S = x + y$ and $P = x \times y$:

$x^2 + y^2 = 34$

$x \times y = 15$

A farmer has harvested $500$ kg of fruit consisting of apples and pears. Knowing that three times the kilograms of apples is exactly equal to twice the kilograms of pears, set up and solve a system of linear equations to determine how many kilograms of each fruit he harvested.

In a chemical laboratory, we wish to obtain $100$ litres of an acid solution at $30\%$. To do so, two solutions available in the storeroom are to be mixed: one with a concentration of $20\%$ and another at $45\%$. Set up and solve a system of linear equations to calculate how many litres of each solution should be used.

Two cyclists set off simultaneously from two towns A and B, separated by a distance of $120$ km, travelling in opposite directions towards each other. The cyclist setting off from A travels at a constant speed $5$ km/h faster than the cyclist setting off from B. Knowing that they pass each other after exactly $3$ hours, set up a system of equations to find the speed of each cyclist and the distance each cyclist has travelled up to the meeting point.