Simultaneous Equations Flashcards

All 16 cards in this deck

What are the steps to solve a pair of linear simultaneous equations by elimination?

e.g. $3x+2y=12$ and $x+2y=8$

1. Match coefficients, then add or subtract to eliminate one unknown: subtracting gives $2x=4$, so $x=2$
2. Substitute back into either equation: $2+2y=8$, so $y=3$
3. State both values: $x=2$, $y=3$

When eliminating, do you add or subtract the two equations?

e.g. $3x+2y=12$ and $3x-2y=6$

Same signs on the matched terms: subtract. Different signs: add.
Here the $y$ terms have different signs, so add.

What are the steps to solve simultaneous equations giving $x$ and $y$ in terms of a constant $k$?

e.g. $y=2kx$ and $3x+4y=k$

1. Substitute one equation into the other to eliminate a variable: $3x+8kx=k$
2. Factorise and divide to get the first unknown: $x=\dfrac{k}{3+8k}$
3. Substitute back for the other: $y=\dfrac{2k^2}{3+8k}$

Graphically, what does the solution of two linear simultaneous equations represent?

The coordinates of the point where the two straight lines intersect.

What are the steps to find unknown constants in a curve's equation from points it passes through?

e.g. $y=ax^2+bx$ passes through $(2,20)$ and $(6,12)$

1. Substitute the first point: $4a+2b=20$
2. Substitute the second point: $36a+6b=12$
3. Solve the pair simultaneously: $a=-2$, $b=14$

True or false? A pair of linear simultaneous equations always has exactly one solution.

False. If the lines are parallel there is no solution; if they are the same line there are infinitely many.

What are the steps to solve a linear and a second-order pair of simultaneous equations?

e.g. $y=x+1$ and $x^2+y^2=25$

1. Rearrange the linear equation for one unknown: $y=x+1$
2. Substitute into the second-order equation and simplify: $x^2+x-12=0$
3. Solve, then substitute each value back: $(3,4)$ and $(-4,-3)$

How many solutions can a pair of one linear and one second-order equation have, and what do they represent?

Up to two pairs of values, each pair being the coordinates of a point of intersection of the line and the curve.

True or false? To solve a linear and a second-order equation, you substitute the second-order equation into the linear one.

False. Rearrange the linear equation and substitute that expression into the second-order equation.

After substituting, the resulting quadratic has no real solutions. What does this tell you about the line and the curve?

They do not intersect — there are no solutions to the simultaneous equations.

True or false? If you find $x=3$ or $x=-4$ and $y=4$ or $y=-3$, any pairing of the values is a solution.

False. Each $x$ must be paired with the $y$ found by substituting that same $x$.

What are the steps to solve three linear equations in three unknowns by elimination?

e.g. $x+y+z=6$, $2x+y-z=1$, $x-y+z=2$

1. Eliminate the same unknown from two different pairs: adding gives $3x+2y=7$ and $3x=3$
2. Solve those two equations in two unknowns: $x=1$, $y=2$
3. Substitute both into an original equation: $z=3$

True or false? Three linear equations in three unknowns may be solved by trial and improvement.

False. Trial and improvement is not accepted; use systematic elimination or substitution.

True or false? When solving three equations in three unknowns, it does not matter which unknown you eliminate first.

True. Any of the three may be eliminated first, as long as you eliminate the same one from both pairs.

After the first elimination step with three equations in three unknowns, what should you have?

Two equations containing the same two unknowns, which are then solved as an ordinary simultaneous pair.

How can you check the three values found from three linear equations in three unknowns?

Substitute all three values into each original equation and confirm every one is satisfied.