Equations and Systems · Equations, Inequalities and Systems
Systems of Linear Equations
At a Cubs game, Marcus pays $\$14$ for 2 hot dogs and 1 lemonade. His friend Ava pays $\$13$ for 1 hot dog and 2 lemonades. What does each item cost? Marcus's receipt alone can't tell you: a $\$5$ hot dog with a $\$4$ lemonade works, but so does a $\$6$ hot dog with a $\$2$ lemonade. Put the two receipts together, though, and only one pair of prices fits both. That is the idea of a system of equations.
Systems and Their Solutions
To check a proposed solution, plug it into both equations. For $(4, 3)$: $4 + 3 = 7$ ✓ and $2(4) - 3 = 5$ ✓. Because both are true, $(4, 3)$ is a solution. If it fails even one equation, it is not a solution of the system.
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Check if the ordered pair $(x, y) = (2, -1)$ is a solution to the following system of linear equations: $3x + 2y = 4$ $x - 4y = 6$
Check if the ordered pair $(x, y) = (2, -1)$ is a solution to the following system of linear equations:
Every solution of one linear equation is a point on its line. A solution of the system has to lie on both lines, so it is the point where the lines cross.
Here are the lines $y = x + 1$ and $y = -2x + 7$:
Two lines in a plane can be related in only three ways, so a system has only three possible outcomes:
Drag the sliders and watch where the two lines meet. Can you make them parallel?
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Look at the graph of two lines that form a system of linear equations. State the solution of the system and classify it according to its number of solutions (consistent…
Look at the graph of two lines that form a system of linear equations. State the solution of the system and classify it according to its number of solutions (consistent independent with a unique solution, consistent dependent with infinitely many solutions, or inconsistent).
Graphing is slow, and it is hard to read exact answers. Algebra gives exact answers. The first algebraic method is substitution: if you know what $y$ equals, you can swap that expression in for $y$ anywhere it appears.
When both equations are already solved for $y$, the right sides must be equal. Some problems say "set the expressions equal to each other." That is just substitution.
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Solve the following system of equations using the substitution method: $x = 2y + 5$ $3x - 4y = 11$
Solve the following system of equations using the substitution method:
Solve the following system of linear equations, simplifying the expressions with fractions and parentheses first: $\frac{3(x - 1)}{2} + \frac{y + 2}{4} = \frac{11}{4}$…
Solve the following system of linear equations, simplifying the expressions with fractions and parentheses first:
Sometimes the algebra "breaks." Try adding $x + y = 3$ and $-x - y = -5$: you get $0 = -2$. That is false for every $x$ and $y$, so no pair works. The lines are parallel.
Now try $2x - 4y = 6$ and $-x + 2y = -3$. Multiply the second by $2$: $-2x + 4y = -6$. Adding gives $0 = 0$. That is always true, so every point on the line works. The two equations describe the same line.
Slide $b$ and watch the second line disappear into the first:
These ratios also let you find a missing coefficient.
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Determine the value of the parameter $k$ so that the following system of equations is inconsistent (that is, the lines representing it are parallel): $kx + 3y = 5$…
Determine the value of the parameter $k$ so that the following system of equations is inconsistent (that is, the lines representing it are parallel):
Determine the value of the parameter $m$ so that the following system has infinitely many solutions (is a consistent dependent system): $3x - my = 12$ $-6x + 8y = -24$
Determine the value of the parameter $m$ so that the following system has infinitely many solutions (is a consistent dependent system):
Look at the following graph of a system formed by two lines, $r_1$ and $r_2$, whose equations are of the form $ax+by=c$ and $a'x+b'y=c'$. From the graph, classify the…
Look at the following graph of a system formed by two lines, $r_1$ and $r_2$, whose equations are of the form $ax+by=c$ and $a'x+b'y=c'$. From the graph, classify the system according to its number of solutions and explain what algebraic relationship must exist between the coefficients of both equations for this situation to occur.
Systems shine in word problems with two unknown quantities. The hard part is turning words into equations.
Words
Algebra
"the sum of two numbers is 54"
$x + y = 54$
"their difference is 12"
$x - y = 12$
"$x$ is twice $y$"
$x = 2y$
"$x$ is 5 more than $y$"
$x = y + 5$
Back to the beginning: the only prices that fit both receipts are $\$5$ for a hot dog and $\$4$ for a lemonade. Marcus's receipt alone allowed many pairs, such as $(5, 4)$ and $(6, 2)$. Ava's receipt kept only $(5, 4)$.
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The sum of two numbers is $54$ and their difference is $12$. Set up a system of linear equations and find both numbers.
The sum of two numbers is $54$ and their difference is $12$. Set up a system of linear equations and find both numbers.
At a coffee shop, we were charged $\$7,50$ for $3$ coffees and $2$ croissants. If we buy $1$ coffee and $4$ croissants of the same type, the price is $\$6,50$. How much…
At a coffee shop, we were charged $\$7,50$ for $3$ coffees and $2$ croissants. If we buy $1$ coffee and $4$ croissants of the same type, the price is $\$6,50$. How much does each coffee and each croissant cost?
In a repair shop, there is a total of $35$ vehicles between cars and motorcycles. If we count all their wheels (excluding spare tires), there is a total of $110$ wheels.…
In a repair shop, there is a total of $35$ vehicles between cars and motorcycles. If we count all their wheels (excluding spare tires), there is a total of $110$ wheels. How many cars and how many motorcycles are in the shop?
Many problems count items that each have a value: tickets, coins, gallons of oil, dollars in a fund. They always give you two equations of different kinds.
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A merchant mixes olive oil at $\$4$/gallon with sunflower oil at $\$2$/gallon to obtain $60$ gallons of a mixture that will be sold at $\$3,50…
A merchant mixes olive oil at $\$4$/gallon with sunflower oil at $\$2$/gallon to obtain $60$ gallons of a mixture that will be sold at $\$3,50$/gallon. Set up a system of equations to find out how many gallons of each type of oil were used.
There are $40$ coins in a piggy bank, some are $\$0,20$ and others are $\$0,50$. If there is a total of $\$12,50…
There are $40$ coins in a piggy bank, some are $\$0,20$ and others are $\$0,50$. If there is a total of $\$12,50$, how many coins of each type does the piggy bank contain?
A person invests a total of $\$25000$ in two different investment funds. The first fund offers an annual interest rate of $4\,\%$, while the second offers $6\,\%…
A person invests a total of $\$25000$ in two different investment funds. The first fund offers an annual interest rate of $4\,\%$, while the second offers $6\,\%$. If after one year the total profit obtained from both funds amounts to $\$1320$, how much money did they invest in each one?
When percents are involved, the value equation tracks the actual amount of the ingredient, or the price after the change.
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A laboratory has two solutions of the same acid, one with a $10\,\%$ concentration and another with a $25\,\%$ concentration. The aim is to obtain $45$ gallons of a new…
A laboratory has two solutions of the same acid, one with a $10\,\%$ concentration and another with a $25\,\%$ concentration. The aim is to obtain $45$ gallons of a new solution with a $16\,\%$ concentration. How many gallons of each original solution must be mixed?
In an electronics store, a laptop and a tablet cost $\$1500$ together. During the sale week, a $15\,\%$ discount is applied to the laptop and a $10\,\%…
In an electronics store, a laptop and a tablet cost $\$1500$ together. During the sale week, a $15\,\%$ discount is applied to the laptop and a $10\,\%$ discount to the tablet, making the combined price $\$1305$. Set up a system of equations to find the original price of each item.
$5$ years ago, a mother's age was three times her son's age. In $10$ years, the mother's age will be twice the son's age. Set up a system of linear equations to…
$5$ years ago, a mother's age was three times her son's age. In $10$ years, the mother's age will be twice the son's age. Set up a system of linear equations to calculate their current ages.
The perimeter of a rectangular plot measures $120$ feet. If the length of the plot is $10$ feet longer than the width, what are the dimensions of the plot?
The perimeter of a rectangular plot measures $120$ feet. If the length of the plot is $10$ feet longer than the width, what are the dimensions of the plot?
Two cyclists leave at the same time from two cities, A and B, which are $78$ miles apart, and travel toward each other in a straight line. They take exactly $3$ hours to…
Two cyclists leave at the same time from two cities, A and B, which are $78$ miles apart, and travel toward each other in a straight line. They take exactly $3$ hours to meet. If the speed of the first cyclist is $4$ mph faster than the second, at what speed is each one traveling?