Equations and Systems · Equations, Inequalities and Systems
Systems of Nonlinear Equations
You have 28 feet of fencing and want a rectangular garden bed with exactly 48 square feet of soil. How long and how wide should it be? Again you have two unknowns and two equations, but this time one equation has a product of the unknowns instead of a sum. That makes the system nonlinear, and we need a slightly different toolkit.
What makes a system nonlinear
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Solve the system formed by a linear and a quadratic equation: $x - y = 2$ $x^2 + y^2 = 34$
Solve the system formed by a linear and a quadratic equation:
Each equation in a system has a graph, and a solution is a point that lies on both graphs. A linear equation graphs as a line. An equation like $y=x^2-4x$ graphs as a parabola. So a line-and-parabola system asks: where do they cross?
If both equations are already solved for $y$, set the two expressions equal to each other. For example, $y=x^2-2x-3$ and $y=x+1$ give $x^2-2x-3=x+1$, so $x^2-3x-4=0$, so $(x-4)(x+1)=0$. Then $x=4$ or $x=-1$, and the points are $(4, 5)$ and $(-1, 0)$.
Move the sliders and watch how the line meets the parabola.
After substituting, you get a quadratic equation, and its discriminant $b^2-4ac$ tells you how many real solutions there are:
Discriminant
Solutions
Picture
$b^2-4ac>0$
two
the line cuts the parabola twice
$b^2-4ac=0$
one
the line just touches the parabola (tangent)
$b^2-4ac<0$
none
the line misses the parabola
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Solve the following system of nonlinear equations: $y = x^2 - 4x$ $y = x - 6$
Solve the following system of nonlinear equations:
Sometimes the nonlinear part is a fraction or a radical. First turn the equation into a polynomial equation, then substitute as usual.
Fractions. Multiply both sides by the LCD to clear denominators. A ratio like $\dfrac{x}{y}=2$ becomes $x=2y$ (with $y\neq0$). For example, with $\dfrac{x}{y}=2$ and $x^2+y^2=20$: substitute $x=2y$ to get $4y^2+y^2=20$, so $y^2=4$ and $y=\pm2$. The solutions are $(4,2)$ and $(-4,-2)$.
Radicals. Isolate the radical, square both sides, and check every answer. Squaring can create answers that do not satisfy the original equation (extraneous solutions).
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Find the solution to the system: $\frac{x}{y} = 3$ $x^2 - y^2 = 32$
A number consists of two digits. The product of these digits is $18$. If $27$ is subtracted from this number, another number is obtained with the same digits but in…
A number consists of two digits. The product of these digits is $18$. If $27$ is subtracted from this number, another number is obtained with the same digits but in reverse order. What is the original number? (Remember that a two-digit number $xy$ is expressed algebraically as $10x + y$).
Geometry: perimeter, area and the Pythagorean Theorem
In geometry problems, the perimeter usually gives a linear equation, while the area (or the Pythagorean Theorem) gives a nonlinear one.
Back to the start: the garden bed should be 6 ft by 8 ft.
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The area of a rectangular carpet is $6 \text{ m}^2$ and its perimeter measures $10 \text{ m}$. Set up a system of nonlinear equations and calculate the dimensions of the…
The area of a rectangular carpet is $6 \text{ m}^2$ and its perimeter measures $10 \text{ m}$. Set up a system of nonlinear equations and calculate the dimensions of the carpet.
The hypotenuse of a right triangle measures $13 \text{ cm}$. It is known that one of the legs measures $7 \text{ cm}$ more than the other. Set up a system of nonlinear…
The hypotenuse of a right triangle measures $13 \text{ cm}$. It is known that one of the legs measures $7 \text{ cm}$ more than the other. Set up a system of nonlinear equations to calculate the length of both legs.
The area of a rhombus is $24 \text{ cm}^2$ and the sum of the lengths of its diagonals is $14 \text{ cm}$. Determine the length of both diagonals by setting up a system…
The area of a rhombus is $24 \text{ cm}^2$ and the sum of the lengths of its diagonals is $14 \text{ cm}$. Determine the length of both diagonals by setting up a system of equations.
Some problems change a shape: one side gets shorter, another longer, or a border is added all around. You then have an original equation and a new equation. Expand the new one and substitute the original to simplify it.
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A rectangular plot has an area of $600 \text{ m}^2$. If its length is reduced by $5 \text{ m}$ and its width is increased by $5 \text{ m}$, the area remains the same.…
A rectangular plot has an area of $600 \text{ m}^2$. If its length is reduced by $5 \text{ m}$ and its width is increased by $5 \text{ m}$, the area remains the same. Calculate the original dimensions of the plot.
A rectangular play area is going to be built in a park, surrounded by an artificial grass path $1 \text{ m}$ wide around its entire perimeter. The area of the play area…
A rectangular play area is going to be built in a park, surrounded by an artificial grass path $1 \text{ m}$ wide around its entire perimeter. The area of the play area must be $48 \text{ m}^2$ and the total area (play area plus path) must be $96 \text{ m}^2$. Set up a system of equations to find the dimensions of the play area.
A closed cylindrical container has a total surface area (lateral area plus the two bases) of $150\pi \text{ cm}^2$. It is also known that the sum of the base radius and…
A closed cylindrical container has a total surface area (lateral area plus the two bases) of $150\pi \text{ cm}^2$. It is also known that the sum of the base radius and the height of the cylinder is $15 \text{ cm}$. Calculate the radius and height of the container. (Remember that the total surface area of a cylinder is $A = 2\pi r^2 + 2\pi rh$).
Many word problems follow one pattern: (number of people) × (price each) = total. When the group changes but the total stays the same, you get a second equation. Distance problems work the same way: rate × time = distance.
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A group of students buys a gift for a classmate for $120$ dollars, to be paid in equal parts. At the last minute, $2$ more students join, which means that each of the…
A group of students buys a gift for a classmate for $120$ dollars, to be paid in equal parts. At the last minute, $2$ more students join, which means that each of the original students pays $3$ dollars less. How many students were there at the beginning and how much did each one pay?
A travel agency organizes a trip for a total cost of $600$ dollars. Shortly before leaving, $5$ people cancel their trip, which forces the rest to pay $4$ dollars more…
A travel agency organizes a trip for a total cost of $600$ dollars. Shortly before leaving, $5$ people cancel their trip, which forces the rest to pay $4$ dollars more each to cover the total cost. How many people finally went on the trip and how much did each person pay?
A cyclist travels a distance of $120 \text{ km}$ at a constant speed. If they had traveled $5 \text{ km/h}$ faster, they would have taken $2$ hours less to complete the…
A cyclist travels a distance of $120 \text{ km}$ at a constant speed. If they had traveled $5 \text{ km/h}$ faster, they would have taken $2$ hours less to complete the same journey. Set up a system of equations to calculate the cyclist's speed and the time taken.
A nonlinear system can give several mathematical solutions, but not all of them fit the situation. It can also give no real solution at all.
When a word problem gives a negative discriminant, double-check your setup first (a misread number is the usual cause). If the setup is right, state clearly that the situation is impossible.
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The product of two positive integers is $156$ and one of them is consecutive to the other. Set up a system of equations and find these numbers.
The product of two positive integers is $156$ and one of them is consecutive to the other. Set up a system of equations and find these numbers.