Exercises: Solving methods

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 non-linear system of equations using the substitution method:

$x - y = 2$

$x^2 + y^2 = 10$

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

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

$x - y = 5$

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:

$\log_2(x + y) = 3$

$x - 2y = 2$

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.

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:

$2x - y + 3z = 1$

$x + 2y - z = 4$

$4x - y + z = 2$

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 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:

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

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

In a chemical laboratory, we wish to obtain $100$ liters 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 liters of each solution should be used.

The perimeter of a rectangular plot is $140$ m and the length of its diagonal is $50$ m. Set up a system of non-linear equations describing the situation and solve it to determine the dimensions of the plot.

A technology company assembles three types of drones: A, B, and C. In total, they manufacture $60$ drones per month. The number of type A drones is double that of type C drones, and the sum of type B and C drones exceeds type A drones by $12$. Set up a system of three equations in three unknowns and solve it using Gaussian elimination.

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

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

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

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 mixed system combining an exponential equation and a logarithmic equation:

$2^x \times 4^y = 256$

$\log_3(x) + \log_3(y) = \log_3(6)$

An investor splits a total capital of $3\,000\,000$ pesos among three investment funds: equities, fixed income, and government bonds. The equities fund offers a $8\%$ annual return, the fixed income fund $5\%$, and the bonds $3\%$. The investor decides to allocate twice as much capital to fixed income as to government bonds. If after one year the total profits generated amount to $174\,000$ pesos, how much money did they invest in each fund? Set up the corresponding system and solve it using Gaussian elimination.

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.

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

$x^2 + y^2 = 34$

$x \times y = 15$

Determine analytically the points of intersection between the circle with equation $x^2 + y^2 = 25$ and the rectangular hyperbola with equation $x \times y = 12$. Set up the associated non-linear system, solve it detailing all steps, and explain how many real solutions exist.