Three adult cinema tickets and two children's tickets add up to 34 €. That sentence translates into $3x + 2y = 34$. Here there are two unknowns and only one equation: there is not a single solution, but a whole range of pairs $(x, y)$ that satisfy it. Let's see what that means and how to work with it.
With just one equation of this kind there are infinitely many solutions: choose a value of $x$ and the value of $y$ follows from it (or the other way round).
To check a solution, substitute both coordinates: $(4, 11)$ works because $3 \cdot 4 + 2 \cdot 11 = 12 + 22 = 34$ ✓. $(3, -1)$ does not, because $3 \cdot 3 + 2 \cdot (-1) = 7 \ne 34$.
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Check whether the pairs of values $(1, 2)$, $(3, -1)$ and $(0, 4)$ are solutions to the linear equation $3x + 2y = 7$.
Check whether the pairs of values $(1, 2)$, $(3, -1)$ and $(0, 4)$ are solutions to the linear equation $3x + 2y = 7$.
State with reasons which of the following equations are linear equations with two unknowns: a) $x^2 + y = 7$ b) $3x - \frac{y}{4} = 1$ c) $x \cdot y = 12$ d)…
State with reasons which of the following equations are linear equations with two unknowns:
If you mark on a plane all the pairs $(x, y)$ that satisfy $ax + by = c$, you get a straight line. That is the reason for the name "linear".
To draw it by hand, two points are enough: the most convenient ones are usually where the line crosses each axis, that is, by setting $x = 0$ and then $y = 0$.
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Given the equation $7x + 3y = 21$, find the value of $y$ when $x = 0$, and the value of $x$ when $y = 0$.
Given the equation $7x + 3y = 21$, find the value of $y$ when $x = 0$, and the value of $x$ when $y = 0$.
Given the linear equation with fractional coefficients $\frac{x - 1}{2} + \frac{2y + 3}{5} = 1$, express it in its general form $ax + by = c$ by clearing the…
Given the linear equation with fractional coefficients $\frac{x - 1}{2} + \frac{2y + 3}{5} = 1$, express it in its general form $ax + by = c$ by clearing the denominators. Then, find a solution in which $x$ is a negative integer.
Decide what each unknown represents and write it down.
Translate the statement word by word: "twice", "the sum of", "costs in total"…
Check that the equation makes sense with a simple value.
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Translate the following statement into algebraic language using a linear equation with two unknowns: "The price of three adult cinema tickets plus the price of two child…
Translate the following statement into algebraic language using a linear equation with two unknowns: "The price of three adult cinema tickets plus the price of two child tickets totals 34 euros".
An exam consists of multiple-choice questions worth 2 points each and open-ended problems worth 5 points each. Set up a linear equation that relates the number of…
An exam consists of multiple-choice questions worth 2 points each and open-ended problems worth 5 points each. Set up a linear equation that relates the number of questions of each type to obtain a maximum score of 30 points. Find a solution in which the number of multiple-choice questions is exactly five times the number of open-ended problems.
A bicycle rental company charges a fixed insurance fee plus an amount for each hour of use. If renting a bicycle for 3 hours costs 14 euros and for 5 hours costs 22…
A bicycle rental company charges a fixed insurance fee plus an amount for each hour of use. If renting a bicycle for 3 hours costs 14 euros and for 5 hours costs 22 euros, set up a linear equation relating the total cost ($y$) to the hours of use ($x$). What is the value of the fixed fee?