True or False? Both equations in a pair must use the same units.

True. The two equations describe the same two unknowns, so a price written in pounds in one and in pence in the other would give values that contradict each other. Either unit works, provided every total in both equations matches it.

National 5MathsSQAFlashcardsAlgebraic SkillsSimultaneous Equations

Simultaneous Equations (SQA National 5 Maths): Flashcards

Exam code: X847 75

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FrontAlgebraic Solution to Simultaneous Equations
Define linear simultaneous equations.

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Cards in this collection (20)

Define linear simultaneous equations.
Linear simultaneous equations are two equations that have to be solved at the same time, to find values that work in both of them.
Linear means there are no terms such as $x^{2}$ or $y^{2}$ .
Why are two equations needed to find two unknowns?
One equation on its own has endless pairs of values that satisfy it, so it cannot pin down both unknowns.
A second equation narrows those pairs down to the one that works in both, which for $3 x + 2 y = 11$ and $2 x - y = 5$ is $x = 3$ and $y = 1$ .
What is the first move when solving simultaneous equations by elimination?
Make the coefficient of one of the variables the same size in both equations, by multiplying every term of each equation as needed.
To eliminate $x$ from $3 x + 2 y = 11$ and $2 x - y = 5$ , multiply the first by $2$ and the second by $3$ to give $6 x$ in each.
True or False?
When the terms you want to eliminate have the same sign, you add the two equations together.
False.
Terms with the same sign are removed by subtracting one equation from the other.
Adding is what removes them when their signs are different, as with $+ 2 y$ in one equation and $- 2 y$ in the other.
Complete the result of subtracting the second equation from the first.
$6 x + 4 y = 22$
$6 x - 3 y = 15$
$______y =______$
The completed subtraction is:
$7 y = 7$
The $y$ terms give $4 y - (- 3 y) = 7 y$ , which is where the negatives most often go wrong.
How does the substitution method solve a pair of simultaneous equations?
Rearrange one equation into $y = \dots$ or $x = \dots$ , then replace that variable throughout the other equation.
Rearranging $2 x - y = 5$ to $y = 2 x - 5$ turns $3 x + 2 y = 11$ into $3 x + 2 (2 x - 5) = 11$ , which gives $x = 3$ .
Once one unknown has been found how do you find the other?
Substitute the value you have into either of the original equations and solve for the remaining unknown.
Putting $x = 3$ into $5 x + 4 y = 13$ gives $15 + 4 y = 13$ , so $y = - \frac{1}{2}$ .
How can you be sure a pair of solutions is correct?
Substitute both values into the equation you did not use to find the second unknown, and check that it balances.
With $x = 3$ and $y = - \frac{1}{2}$ in $4 x - 6 y = 15$ , the left-hand side comes to $12 + 3 = 15$ .
Where does a graph show the solution to a pair of simultaneous equations?
The solution is at the point of intersection, the place where the two lines cross.
The $x$ and $y$ coordinates of that point are the $x$ and $y$ solutions of the equations.
How do you solve simultaneous equations graphically when no graph is given?
Plot both equations yourself on the same set of axes, using a table of values or by rearranging each one into $y = m x + c$ first.
Both lines have to be drawn accurately enough for the crossing point to be read off.
True or False?
A single point on the graph gives both the $x$ solution and the $y$ solution.
True.
The point of intersection is the only point lying on both lines, so its two coordinates satisfy both equations at once.
For $2 x - y = 3$ and $3 x + y = 7$ that point is $(2, 1)$ , giving $x = 2$ and $y = 1$ .
Why is it useful to rearrange each equation into $y = m x + c$ before plotting?
That form gives the gradient and the crossing point on the $y$ axis directly, which is enough to draw each line without a table of values.
Rearranging $2 x - y = 3$ gives $y = 2 x - 3$ , a line through $(0, - 3)$ with gradient $2$ .
How does a table of values help you plot a straight line?
Choosing a few $x$ values and working out the matching $y$ values gives a set of points to plot.
Joining those points with a ruler produces the line, whatever form the equation was given in.
Two lines drawn on the same axes cross at $(4, - 3)$ so complete the solution they give.
$x =______and y =______$
The completed solution is:
$x = 4 and y = - 3$
Coordinates are always written with $x$ first, so the crossing point translates straight into the two solutions.
What is the first thing to do when a problem describes two unknowns?
Introduce a letter for each unknown and write down exactly what it stands for, including any units.
For a question about prices, $x$ might be the price of an apple in pence and $y$ the price of a banana in pence.
How does the fact that 3 apples and 2 bananas cost 180p become an equation?
Each quantity multiplies the letter standing for the price of that item, and the total goes on the other side of the equals sign.
With $x$ pence for an apple and $y$ pence for a banana that gives $3 x + 2 y = 180$ .
True or False?
Both equations in a pair must use the same units.
True.
The two equations describe the same two unknowns, so a price written in pounds in one and in pence in the other would give values that contradict each other.
Either unit works, provided every total in both equations matches it.
Six bagels and twelve chocolate bites cost nine pounds, so complete the equation using $b$ and $c$ for the two prices in pounds.
$______b +______c = 9$
The completed equation is:
$6 b + 12 c = 9$
Both letters stand for the price of one item, which is why the numbers of items become the coefficients.
Why is finding $x$ and $y$ not always the end of the question?
The letters were only ever a way of describing the situation, so the answer has to be put back into that situation with its units.
Two numbers with a sum of $19$ and a difference of $5$ give $x = 12$ and $y = 7$ , but a question asking for their product wants $84$ .
Do you always have to use $x$ and $y$ as the two unknowns?
No, any pair of letters works, and letters that match the situation often make the working easier to follow.
Using $b$ for the price of a bagel and $c$ for the price of a chocolate bite is clearer than $x$ and $y$ .

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