Solution Methods
At the street market, Ximena buys 3 tortas (sandwiches) and 2 sodas and pays $\$190$. Diego buys 2 tortas and 3 sodas and pays $\$170$. How much does each item cost? These are two equations with two unknowns. The idea of this whole section is just one: turn the system into something you already know how to solve. First we will see the five methods for $2\times 2$ linear systems and how to choose among them. Then we will use them in problems. At the end there are extension sections with more complicated systems.
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:
$3x - 2y = 7$
$5x + 4y = 8$
Graphing method
Each linear equation with two unknowns is a line in the coordinate plane. The solution of the system is the point where the two lines cross, because that point satisfies both equations at once.
In the interactive, the lines $y=-x+5$ and $y=x-1$ cross at $(3, 2)$. Check: $-3+5=2$ and $3-1=2$ ✓.
Depending on how the lines are positioned, there are three cases:
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They cross at one point: unique solution (consistent and independent system).
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They are parallel: no solution (inconsistent).
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They coincide: infinitely many solutions (consistent and dependent).
Determinants (Cramer's rule)
There is a method that does not solve for anything: it uses numbers calculated from the coefficients. For a system $\begin{cases} ax+by=e \\ cx+dy=f \end{cases}$ we define the determinant $\begin{vmatrix} a & b \\ c & d \end{vmatrix}=ad-bc$.
If $D=0$ you cannot divide: the lines are parallel or coincide, so there is no solution or there are infinitely many.
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 $\lambda$.
Which method should I choose?
There is no "best" method: there is one that is more convenient for each system.
| If the system has… | Use |
|---|---|
| an unknown already solved for or with coefficient $1$ | substitution |
| the same unknown easily solved for in both equations | equating |
| equal or opposite coefficients (or easy to make equal) | elimination |
| a parameter, or you want to know how many solutions it has | determinants |
| equations that are already lines and you just want to see the crossing | graphing |
Problems with two unknowns
A good setup has three steps: (1) choose unknowns with their meaning and units; (2) translate each piece of data into an equation; (3) solve, check, and answer in words.
| Situation | Typical equations |
|---|---|
| Two quantities with a total and a relationship | $x+y=\text{total}$, $ax=by$ |
| Mixtures | total amount and amount of pure substance |
| Motion | $d=vt$; if they approach each other, $d_1+d_2=D$ |
| Prices | $\text{items}\times\text{price}=\text{total}$ |
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.
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.
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: a torta costs $\$46$ and a soda $\$26$, as Cramer's rule showed.
Extension: nonlinear systems
A system is nonlinear if some equation has terms like $x^2$, $xy$, or $y^2$. The methods are the same, but you almost always end up with a quadratic equation and there are several solutions.
When both equations have the same squared terms, it is convenient to add or subtract as in elimination.
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.
Extension: sum and product
If the system does not change when you swap $x$ and $y$ (it is symmetric), it helps to use $S=x+y$ and $P=xy$. Remember that $x^2+y^2=(x+y)^2-2xy=S^2-2P$. And if you know $S$ and $P$, then $x$ and $y$ are the roots of $t^2-St+P=0$.
Another way is to substitute $y=\dfrac{k}{x}$ into the other equation. You get an equation in $x^4$ that is solved as a quadratic in $x^2$. This is what happens when you intersect a circle with a hyperbola $xy=k$.
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.
Extension: exponentials and logarithms
Case 1: the bases can be made equal. You write both sides as powers of the same base and set the exponents equal.
Case 2: different bases with exponents that differ by constants. We use $u=2^x$ and $v=3^y$ and the properties $a^{x+1}=a\cdot a^x$, $a^{y-1}=\dfrac{a^y}{a}$.
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:
$2^{x+y} = 32$
$3^{x-y} = 9$
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 $v = 2^y$:
$3^x + 2^{y+1} = 17$
$3^{x-1} - 2^y = 1$
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:
$2^x \times 4^y = 256$
$\log_3(x) + \log_3(y) = \log_3(6)$
Logarithms
We use the definition $\log_a b=c\Leftrightarrow a^c=b$ and the properties $\log(AB)=\log A+\log B$, $\log\dfrac{A}{B}=\log A-\log B$ and $\log A^n=n\log A$.
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:
$\log_2(x + y) = 3$
$x - 2y = 2$
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:
$\log(x) + \log(y) = 2$
$\log(x^2) - \log(y) = 1$
Extension: radicals and algebraic fractions
Radicals. You solve for one unknown, substitute, isolate the root, and square. Squaring can create false solutions, which is why you check at the end.
Fractions. If each unknown appears only in denominators, we use $u=\dfrac{1}{x}$ and $v=\dfrac{1}{y}$. Before starting, write down the restrictions: $x\neq 0$, $y\neq 0$.
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:
$\frac{2}{x} + \frac{3}{y} = 2$
$\frac{4}{x} - \frac{1}{y} = \frac{1}{6}$
Extension: three unknowns and Gauss's method
With three equations and three unknowns, substitution becomes long. Gauss's method uses elimination in an orderly way until you get an echelon system, which is solved from the bottom up.
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:
$2x - y + 3z = 1$
$x + 2y - z = 4$
$4x - y + z = 2$
Three types of result
When you reduce to echelon form you can find:
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An equation like $0=5$: the system is inconsistent (no solution).
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An equation like $0=0$ (fewer useful equations remain than unknowns): consistent and dependent, with infinitely many solutions. You call one unknown $\lambda$ and express the others in terms of it.
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As many useful equations as unknowns: consistent and independent, with a unique solution.
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 with three unknowns
They are set up the same way as with two, but you need three independent equations: one for each piece of data in the problem.
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.
An investor splits a total capital of $1\,000\,000$ pesos among three investment funds: equities, fixed income, and government bonds. The equities fund offers a $8\%$…
An investor splits a total capital of $1\,000\,000$ 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 $58\,000$ pesos, how much money did they invest in each fund? Set up the corresponding system and solve it using Gaussian elimination.