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:

$x + y + z = 6$

$x - y + 2z = 5$

$x + y - z = 0$

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.

Observe the following system of exponential equations:

Two students suggest different ways to tackle it:

• Aoife suggests solving directly for $3^y$ in the first equation ($3^y = 2^{x+1} - 1$) and substituting that expression into the second equation, using the fact that $3^{y+1} = 3 \times 3^y$.

• Cian suggests first applying the properties of powers to rewrite the system as $2 \times 2^x - 3^y = 1$ and $2^x + 3 \times 3^y = 11$, introducing the auxiliary variables $u = 2^x$ and $v = 3^y$, and solving the resulting linear system by elimination.

1. a) Compare both procedures by analysing their algebraic difficulty. Which of the two methods minimizes the risk of making errors with the properties of powers? Justify your answer.
2. b) Solve the system using the method you consider most convenient and determine the values of $x$ and $y$.

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 $100$ 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.

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:

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

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

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.

A delivery logistics company in the metropolitan area of Dublin decides to renew its fleet by purchasing electric vans and plug-in hybrid vans under the SEAI grant scheme. In a first phase, it acquires $3$ electric vans and $4$ hybrid vans for a total amount of $230\,000$ €. In a second phase, taking advantage of a promotional campaign in which the manufacturer discounts the price of each electric van by $10\,\%$ and each hybrid van by $5\,\%$, the company buys $2$ electric and $5$ hybrid vans for an expenditure of $202\,625$ €.

1. a) Model the situation by setting up a system of two linear equations with two unknowns. Precisely define what each unknown represents and specify its units.
2. b) Solve the system using the method you consider most suitable (substitution, equating, or elimination) to determine the original price of each type of vehicle before the discounts.
3. c) If the company has a remaining budget of $105\,000$ €, will it have enough to purchase one electric van and two hybrid vans at the discounted prices of the second phase? Quantitatively justify your answer.

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 $50000$ € 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 $2900$ €, 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 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.

Design a system of two equations with two unknowns ($x$ and $y$) that simultaneously satisfies the following four conditions:

1. One of the equations must be linear with integer coefficients.

2. The other equation must contain at least one radical where the unknown $x$ is inside the radicand, so that it is not a linear system.

3. When solving the system by isolating and squaring to eliminate the square root, a quadratic equation must be generated whose algebraic solutions are the pairs $(4, 3)$ and $(9, -2)$.

4. The pair $(4, 3)$ must be the only valid real solution to the original system, while the pair $(9, -2)$ must be an extraneous solution generated exclusively by having squared.

Show the step-by-step construction of both equations, solve the complete system you created justifying where and why the extraneous solution appears, and reason why squaring is not a strict equivalence transformation of equations.

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.