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 expressions with radicals:

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

$x - y = 5$

Solve the following system of exponential equations by expressing both sides of each equation as powers of 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 $500$ pounds of fruit between apples and pears. Knowing that three times the pounds of apples is exactly equal to twice the pounds of pears, set up and solve a system of linear equations to determine how many pounds of each fruit they have harvested.

Apply Gaussian elimination to transform and solve the following system of three linear equations with 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 indicate if it is a consistent and determinate system:

$2x - y + 3z = 1$

$x + 2y - z = 4$

$4x - y + z = 2$

Find all the 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 by making 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 value of $x$ and $y$ in the following system of logarithmic equations, previously applying the properties of logarithms to simplify the expressions:

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

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

In a chemical laboratory, it is desired to obtain $100$ liters of an acid solution at $30\%$. To do this, two solutions available in the warehouse will be mixed: one at $20\%$ concentration and another at $45\%$. Set up and solve a system of linear equations to calculate how many liters of each solution must be used.

The perimeter of a rectangular plot measures $140$ ft and the length of its diagonal is $50$ ft. Set up a non-linear system of equations that describes 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 twice that of type C, and the sum of type B and C drones exceeds type A drones by $12$. Set up a system of three equations with three unknowns and solve it using Gaussian elimination.

Solve the following system of equations containing algebraic fractions, clearly indicating the prior 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 in terms of the parameter $k$:

$x + 2y = 5$

$3x + ky = 15$

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

Solve the following system of three equations with three unknowns using Gaussian elimination. Verify that it is a consistent and indeterminate 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 the transformations of Gaussian elimination, 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:

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

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

An investor distributes a total capital of $\$50000$ among three investment funds: equities, fixed income, and government bonds. The equities fund offers an annual return of $8\%$, 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 $\$2900$, 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 cities A and B, separated by a distance of $120$ miles, and travel in opposite directions to meet. The cyclist departing from A travels at a constant speed that exceeds that of the cyclist departing from B by $5$ mph. Knowing that they cross paths exactly after $3$ hours, set up a system of equations to find the speed of each one and the distance each cyclist has travelled to the meeting point.

Solve the following symmetric non-linear system of equations. To facilitate the resolution, use the change of variables $S = x + y$ and $P = x \cdot y$:

$x^2 + y^2 = 34$

$x \cdot y = 15$

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