What do you need in order to set up a pair of simultaneous equations?

Two letters, one for each unknown, and two different equations connecting them. The two equations have to say genuinely different things about the same two unknowns.

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Simultaneous Equations (Cambridge (CIE) IGCSE Maths: Core): Flashcards

Exam code: 0580 & 0980

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FrontSolving Simultaneous Equations
Define linear simultaneous equations.

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Cards in this collection (14)

Define linear simultaneous equations.
Linear simultaneous equations are two equations in two unknowns that are solved together, so that the solution satisfies both at the same time.
They are linear because they contain no $x^{2}$ or $y^{2}$ terms.
True or False?
There is more than one pair of values of $x$ and $y$ that satisfies $3 x + 2 y = 11$ .
True.
Both $x = 3, y = 1$ and $x = 1, y = 4$ satisfy it, and there are many more pairs that do.
This is why a second equation is needed to pin the answer down to one pair.
In $3 x + 2 y = 11$ and $2 x - y = 5$ , how do you make the $y$ terms match so that one can be eliminated?
Multiply every term of the second equation by $2$ , which gives $4 x - 2 y = 10$ .
The first equation is left alone, because its $y$ term is already $2 y$ .
Complete the rule for eliminating a term:
When the signs in front of the terms you want to eliminate are the same, $______$ the equations.
When the signs are different, $______$ the equations.
The completed rule is:
When the signs in front of the terms you want to eliminate are the same, subtract the equations.
When the signs are different, add the equations.
You have eliminated $x$ from a pair of simultaneous equations and found that $y = 1$ . How do you find $x$ ?
Substitute $y = 1$ back into one of the original equations and solve it for $x$ .
Either original equation will do, and the other can then be used to check that both values are correct.
True or False?
Subtracting $6 x - 3 y = 15$ from $6 x + 4 y = 22$ gives $y = 7$ .
False.
The $y$ terms give $4 y - (- 3 y) = 7 y$ , not $y$ , because subtracting a negative term adds it on.
The result is $7 y = 7$ , so $y = 1$ .
How do you start solving a pair of simultaneous equations by substitution instead of elimination?
Rearrange one of the equations into the form $y = \dots$ (or $x = \dots$ ), then replace every $y$ in the other equation with that expression.
Put the expression in brackets as you substitute it, so that it is treated as a single quantity.
How do you find the solution of two simultaneous equations from their graphs?
Plot both equations on the same axes and find the point where the two lines intersect.
The $x$ -coordinate of that point is the solution for $x$ , and the $y$ -coordinate is the solution for $y$ .
What do you need in order to set up a pair of simultaneous equations?
Two letters, one for each unknown, and two different equations connecting them.
The two equations have to say genuinely different things about the same two unknowns.
Five apples and one banana cost $230$ cents. Complete the equation, taking $x$ as the price of an apple and $y$ as the price of a banana, both in cents.
$5 x +______= 230$
The completed equation is:
$5 x + y = 230$
The single banana contributes $y$ on its own, with no number written in front of it.
Why must you write down exactly what each letter stands for, including its units?
Because a letter with no stated meaning cannot be interpreted at the end, and a price could be read as either dollars or cents.
In the apple and banana example $x$ and $y$ are prices in cents, which is why the totals are $180$ and $230$ .
True or False?
Once you have found the two unknowns, the problem is always finished.
False.
The question may ask for something built out of them, such as the product of two numbers whose sum is $19$ and whose difference is $5$ .
Those numbers are $12$ and $7$ , so the answer wanted is $84$ .
Solving a problem about apple and banana prices in cents gives $x = 40$ and $y = 30$ . What must you still do?
State what those numbers mean in the context, with their units: an apple costs $40$ cents and a banana costs $30$ cents.
A bare pair of numbers does not answer a question that was asked in words.
True or False?
A money problem can be set up in dollars or in cents, whichever you prefer.
True.
Either works, provided every number in both equations uses the same unit.
In dollars the apple and banana equations become $3 x + 2 y = 1.8$ and $5 x + y = 2.3$ , giving $x = 0.4$ and $y = 0.3$ .

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