Methods for Solving Systems
Two friends, Yaritza and Wilkin, pay at a corner store: together they buy 5 sodas and 3 small bags of bread, and their bills don't match. To find the price of each item you need a system of equations. In this section you review which method to use depending on what the system looks like. Afterward you'll see some harder cases: second-degree systems, exponents, logarithms, radicals, or three unknowns.
Review: Substitution, Equating, and Elimination
Solve the following system of linear equations using the elimination method: $3x - 2y = 7$ $5x + 4y = 8$
Solve the following system of linear equations using the elimination method:
$3x - 2y = 7$
$5x + 4y = 8$
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…
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.
Choosing the Method Based on the Form of the System
There is no "best" method: there is just the one that is most convenient for each system. If one unknown is already almost solved for, use substitution. If the same unknown can be solved for in both equations, use equating. If the coefficients are similar, use elimination.
With nonlinear systems, substitution is usually best: you solve in the linear equation and substitute into the second-degree one.
You can also add or subtract equations when only $x^2$ and $y^2$ appear. For example, in $\begin{cases} x^2+y^2=13 \\ x^2-y^2=5 \end{cases}$, adding gives $2x^2=18$, that is, $x=\pm3$, and then $y=\pm2$: there are four solutions.
Solve the following non-linear system of equations using the substitution method: $x - y = 2$ $x^2 + y^2 = 10$
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: $x^2 + y^2 - 4x = 5$ $x^2 - y^2 - 4x = -3$
Find all solutions to the following non-linear system of equations, in which both equations are quadratic:
$x^2 + y^2 - 4x = 5$
$x^2 - y^2 - 4x = -3$
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…
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.
Symmetric Systems: Sum and Product
When the system is symmetric (if you swap $x$ and $y$ it doesn't change), it's convenient to use $S=x+y$ and $P=xy$. The key is this identity: $x^2+y^2=(x+y)^2-2xy=S^2-2P$.
Solve the following symmetric non-linear system of equations. To make it easier to solve, use the change of variables $S = x + y$ and $P = x \times y$: $x^2 + y^2 = 34$…
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…
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.
Equating Bases and Using the Definition of a Logarithm
If an unknown is in an exponent, write both sides as powers of the same base and set the exponents equal. If it is inside a logarithm, use the definition: $\log_a M=c \Leftrightarrow M=a^c$.
Solve the following exponential system of equations by expressing both sides of each equation as powers with the same base: $2^{x+y} = 32$ $3^{x-y} = 9$
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: $\log_2(x + y) = 3$ $x - 2y = 2$
Apply the definition of a logarithm to set up an equivalent linear system and solve the following system of equations:
$\log_2(x + y) = 3$
$x - 2y = 2$
Change of Variable and Properties of Logarithms
When there are several logarithms, apply their properties first: $\log a+\log b=\log(ab)$, $\log a-\log b=\log\dfrac{a}{b}$, $\log a^n=n\log a$.
Solve the following system of exponential equations using the changes of variable $u = 3^x$ and $v = 2^y$: $3^x + 2^{y+1} = 17$ $3^{x-1} - 2^y = 1$
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:…
Find the values of $x$ and $y$ in the following system of logarithmic equations, applying logarithmic properties beforehand to simplify the expressions:
$\log(x) + \log(y) = 2$
$\log(x^2) - \log(y) = 1$
Solve the following mixed system combining an exponential equation and a logarithmic equation: $2^x \times 4^y = 256$ $\log_3(x) + \log_3(y) = \log_3(6)$
Solve the following mixed system combining an exponential equation and a logarithmic equation:
$2^x \times 4^y = 256$
$\log_3(x) + \log_3(y) = \log_3(6)$
Radicals and Algebraic Fractions
If each unknown appears only in denominators, let $u=\dfrac1x$, $v=\dfrac1y$: the system becomes linear again. First, note the restrictions: $x\neq0$, $y\neq0$.
Find the solution to the following system of equations involving radical expressions: $\sqrt{x} + \sqrt{y} = 5$ $x - y = 5$
Find the solution to the following system of equations involving radical expressions:
$\sqrt{x} + \sqrt{y} = 5$
$x - y = 5$
Solve the following system of equations containing algebraic fractions, clearly stating the initial restrictions on the unknowns: $\frac{2}{x} + \frac{3}{y} = 2$…
Solve the following system of equations containing algebraic fractions, clearly stating the initial restrictions on the unknowns:
$\frac{2}{x} + \frac{3}{y} = 2$
$\frac{4}{x} - \frac{1}{y} = \frac{1}{6}$
Three Unknowns: Gauss's Method
With three equations and three unknowns, substituting or equating becomes slow. The orderly approach is to apply elimination to remove unknowns one by one until you reach a triangular (echelon) system, which is solved from the bottom up. That is Gauss's method.
The allowed transformations (they don't change the solutions): swap two equations, multiply an equation by a nonzero number, and add to one equation a multiple of another.
Find the value of the unknowns $x$, $y$, $z$ in the following system of linear equations, which is already in echelon form: $2x - y + 3z = 14$ $3y - z = 5$ $2z = 8$
Find the value of the unknowns $x$, $y$, $z$ in the following system of linear equations, which is already in echelon form:
$2x - y + 3z = 14$
$3y - z = 5$
$2z = 8$
Apply Gaussian elimination to transform and solve the following system of three linear equations in three unknowns: $x + y + z = 6$ $x - y + 2z = 5$ $x + y - z = 0$
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: $2x - y + 3z = 1$ $x + 2y - z = 4$…
Solve the following system of linear equations using Gaussian elimination and state whether it is a consistent independent system:
$2x - y + 3z = 1$
$x + 2y - z = 4$
$4x - y + z = 2$
What Can Happen When Reducing: Types of Systems
When you reduce a system to triangular form, three things can happen:
| Result when reducing | Type of system |
|---|---|
| Each unknown can be solved for in its own equation | Consistent and independent (one solution) |
| $0=0$ appears | Consistent and dependent (infinitely many solutions, with parameter $\lambda$) |
| $0=k$ with $k\neq0$ appears | Inconsistent (no solution) |
With a parameter in a 2×2 system, you discuss it the same way: if the coefficient becomes zero when you eliminate an unknown, see whether you are left with $0=0$ (infinitely many solutions) or $0=k$ (none). In $\begin{cases} x+3y=4 \\ 2x+ky=8 \end{cases}$, subtracting $2E_1$ from $E_2$ gives $(k-6)y=0$; if $k=6$ there are infinitely many solutions, with $y=\lambda$ and $x=4-3\lambda$. If $k\neq6$, $y=0$ and $x=4$.
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…
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…
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: $x + y + z = 4$ $2x - y + 3z = 5$…
Analyse the following system of linear equations and prove, using Gaussian elimination steps, that it is an inconsistent system:
$x + y + z = 4$
$2x - y + 3z = 5$
$3x + 4z = 10$
Problems Solved with Systems
To set up a problem: (1) name the unknowns, (2) translate each sentence of the text into an equation, (3) solve, (4) check against the statement.
In motion problems, if two cyclists ride toward each other, the distances add up to the initial separation: $d_A+d_B=D$, with $d=v\,t$.
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…
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.
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…
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.
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…
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.
Back to the beginning: with the price of a soda as $x$ and the price of a small bag of bread as $y$, each of Yaritza's and Wilkin's bills is an equation, and any of the methods on this page gives you both prices.