Simultaneous Equations Flashcards

All 12 cards in this deck

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

e.g. $2x + y = 7$ and $3x - y = 8$

1. Match the coefficients of one variable: $y$ and $-y$ already match.
2. Add (or subtract) the equations to eliminate it: $5x = 15$, so $x = 3$.
3. Substitute back into one original equation: $y = 1$.

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

e.g. terms $+4y$ and $-4y$

Add when the matching terms have opposite signs; subtract when they have the same sign.
Here $+4y$ and $-4y$: add.

What are the steps to solve linear simultaneous equations by substitution?

e.g. $y = 2x + 1$ and $3x + y = 11$

1. Substitute the expression into the other equation: $3x + (2x + 1) = 11$.
2. Solve for that variable: $5x = 10$, so $x = 2$.
3. Substitute back to find the other: $y = 5$.

What must you do first when a simultaneous equations question is given in words?

Define a letter for each unknown and form two equations from the information.
e.g. 3 teas and 2 coffees cost £7.80 gives $3t + 2c = 7.80$.

True or false? If two linear equations give parallel lines with different intercepts, the simultaneous equations have no solution.

True. Parallel lines never intersect, so there is no pair of values satisfying both.

What are the steps to solve a linear and a quadratic equation simultaneously?

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

1. Substitute the linear equation into the quadratic: $x^2 + 3 = 4x$.
2. Rearrange to $= 0$ and solve: $x^2 - 4x + 3 = 0$, so $x = 1$ or $x = 3$.
3. Substitute each $x$ into the linear equation: $x = 1, y = 4$ and $x = 3, y = 12$.

For a linear/quadratic pair of simultaneous equations, which equation do you rearrange and which do you substitute into?

Rearrange the linear equation to make one variable the subject, then substitute it into the quadratic equation.

What are the steps to solve a circle equation and a straight line simultaneously?

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

1. Substitute the line into the circle: $x^2 + (x+1)^2 = 25$.
2. Expand and form a quadratic $= 0$, then solve: $x^2 + x - 12 = 0$, so $x = 3$ or $x = -4$.
3. Substitute each $x$ into the line: $x = 3, y = 4$ and $x = -4, y = -3$.

True or false? A straight line and a circle always meet at two points.

False. The line may be a tangent (one point) or miss the circle completely (no solutions).

When solving a line and a curve simultaneously, what does it mean if the resulting quadratic has a repeated root?

The line and curve meet at exactly one point, so the line is a tangent to the curve.

What do the coordinates of the point where two straight-line graphs cross tell you?

e.g. the graphs of $y = -2x$ and $5x - 9y = -46$ cross at $(-2, 4)$

The solution of the two equations solved simultaneously: $x = -2$, $y = 4$.

What are the steps to estimate the solutions of a circle equation and a linear equation using a given graph of the circle?

e.g. the graph of $x^2 + y^2 = 25$ is drawn; solve it with $y = x + 1$

1. Draw the straight line on the same axes: line through $(0, 1)$ with gradient $1$.
2. Mark every point where the line crosses the circle: $(3, 4)$ and $(-4, -3)$.
3. Read off each pair of coordinates as estimates: $x = 3, y = 4$ and $x = -4, y = -3$.