Simultaneous equations

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What simultaneous equations are

Simultaneous equations are two or more equations that share the same unknowns; the solution satisfies all of them at once.
Graphically, each linear equation is a line, and the solution is the point of intersection.
Intersecting non-parallel lines have one solution. Distinct parallel lines have none; equations of the same line have infinitely many.

Elimination

Make the coefficients of one unknown the same (multiply an equation if needed), then add or subtract to eliminate it.
Example: 2x + y = 11 and x - y = 1. Add them: 3x = 12, so x = 4, then y = 3.
If the signs are the same, subtract; if opposite, add.

One point satisfies both equations

Substitution

Rearrange one equation to make a variable the subject, then substitute into the other.
Example: y = 2x + 1 and 3x + y = 11. Substitute: 3x + 2x + 1 = 11, so x = 2, y = 5.

Linear and quadratic pairs

When one equation is quadratic (for example y = x² and y = x + 2), use substitution: x² = x + 2 gives (x - 2)(x + 1) = 0, so there are two solutions: (2, 4) and (-1, 1).
A line and a circle can meet at two points, one (a tangent) or none.

A line and a parabola

Checking

Always substitute your answers into both original equations.

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Practice questions

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1.When two simultaneous linear equations have one unique solution, what does it represent graphically?
Easy
- Athe gradient
- Bthe point of intersection
- Cthe y-intercept
- Dthe midpoint
2.Solve x + y = 7 and x - y = 3.
Easy
- Ax = 5, y = 2
- Bx = 2, y = 5
- Cx = 4, y = 3
- Dx = 7, y = 0
3.Solve 2x + y = 9 and x - y = 0.
Easy
- Ax = 3, y = 3
- Bx = 4, y = 1
- Cx = 2, y = 5
- Dx = 9, y = 9
4.Solve 3x + 2y = 12 and x + 2y = 8.
Easy
- Ax = 2, y = 3
- Bx = 3, y = 1.5
- Cx = 4, y = 0
- Dx = 1, y = 3.5
5.Solve y = x + 2 and 2x + y = 11.
Easy
- Ax = 3, y = 5
- Bx = 4, y = 6
- Cx = 2, y = 4
- Dx = 5, y = 7
6.Solve y = 3x and x + y = 8.
Easy
- Ax = 2, y = 6
- Bx = 3, y = 5
- Cx = 4, y = 4
- Dx = 1, y = 7
7.Solve 2x + 3y = 13 and x - y = 1.
Medium
- Ax = 3.2, y = 2.2
- Bx = 4, y = 1
- Cx = 2, y = 3
- Dx = 5, y = 1
8.Solve 3x + 2y = 16 and 5x - 2y = 8.
Medium
- Ax = 3, y = 3.5
- Bx = 2, y = 5
- Cx = 4, y = 2
- Dx = 3, y = 3

More questions

9.Solve 4x + y = 14 and 2x + 3y = 12.
Medium
- Ax = 3, y = 2
- Bx = 2, y = 6
- Cx = 3, y = 3
- Dx = 4, y = -2
10.Solve 3x + 2y = 12 and 2x - y = 1.
Medium
- Ax = 2, y = 3
- Bx = 3, y = 1.5
- Cx = 1, y = 4.5
- Dx = 4, y = 0
11.Two pens and a pencil cost £3.50; a pen and two pencils cost £2.50. What does a pen cost?
Medium
- A£1.50
- B£1.00
- C£0.50
- D£2.00
12.A line y = x + 1 meets y = x² - 1. What are the x-values at the intersections?
Medium
- Ax = 2, -1
- Bx = 1, -2
- Cx = 2, 1
- Dx = 0, 1
13.Solve y = x² and y = 3x - 2.
Medium
- A(1, 1) and (2, 4)
- B(1, 1) only
- C(-1, 1) and (2, 4)
- D(0, 0) and (3, 9)
14.Solve x² + y² = 25 and y = x + 1.
Medium
- A(3, 4) and (-4, -3)
- B(4, 5) and (-5, -4)
- C(3, 4) only
- D(0, 1) and (1, 2)
15.Which method eliminates y in 2x + y = 5 and x - y = 1?
Easy
- Aadd the equations
- Bsubtract the equations
- Cmultiply the first by x
- Ddivide the second by y
16.In 5x + 2y = 11 and 3x + 2y = 7, which operation eliminates y?
Easy
- Aadd
- Bsubtract
- Cmultiply
- Ddivide
17.To eliminate y from 2x + 3y = 7 and 5x + y = 9, you could multiply the second equation by 3 and then:
Medium
- Asubtract the first equation
- Badd the first equation
- Cdivide by x
- Dsquare both sides
18.After finding x you should:
Easy
- Astop
- Bsubstitute to find y
- Cdraw a graph
- Dsquare it
19.If two equations represent distinct parallel lines, how many solutions are there?
Medium
- A0
- B1
- C2
- Dinfinitely many
20.A line and a circle have no common points. The simultaneous equations have:
Medium
- Atwo solutions
- Bone solution
- Cno real solutions
- Dinfinite solutions
21.Check: do x = 2, y = 3 satisfy x + y = 5 and 2x - y = 1?
Easy
- Ayes
- Bno, first fails
- Cno, second fails
- Dneither
22.The solution to x + y = 5 and x - y = 1 is the point:
Medium
- A(3, 2)
- B(2, 3)
- C(5, 1)
- D(4, 1)
23.Simultaneous equations are solved by finding values that satisfy both equations.
Easy
True or false?
24.Substitution means replacing one variable with an expression from another equation.
Easy
True or false?
25.You should always add the equations to eliminate a variable.
Easy
True or false?
26.A line and a parabola can intersect at two points.
Easy
True or false?
27.Two distinct parallel lines give exactly one solution to their simultaneous equations.
Medium
True or false?
28.Which values belong to the solution of x + y = 5 and x − y = 1?
Medium
- Ax = 3
- By = 2
- Cx = 2
- Dy = 3
- Ex = 4
29.Which of these are valid methods?
Medium
- Aelimination
- Bsubstitution
- Cgraphing
- Dguessing and never checking
- Edividing the equations by zero
30.Solve x + y = 9 and x = 4.
Easy
- Ay = 5
- By = 13
- Cy = 4
- Dy = 36
31.Solve y = 2x and y = x + 5.
Easy
- Ax = 5, y = 10
- Bx = 2, y = 4
- Cx = 3, y = 6
- Dx = 10, y = 20
32.Solve 2x + y = 11 and x - y = 1 and show that your answer is correct.
Medium

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