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Worksheet: Solving Systems of Equations Graphically

In this worksheet, we will practice solving a system of two linear equations by considering their graphs and identifying the point of intersection.

Q1:

Find the solution set of the two equations represented by the 𝐿 1 and 𝐿 2 .

  • A { ( 2 , 2 ) }
  • B { ( 3 , 2 ) }
  • C { ( 3 , 3 ) }
  • D { ( 2 , 3 ) }
  • E { ( 2 , 0 ) }

Q2:

Use the shown graph to solve the given simultaneous equations.

  • A π‘₯ = 1 . 5 and 𝑦 = 3
  • B π‘₯ = 0 and 𝑦 = 3
  • C π‘₯ = 1 and 𝑦 = βˆ’ 2
  • D π‘₯ = 1 and 𝑦 = 1
  • E π‘₯ = βˆ’ 2 and 𝑦 = 3

Q3:

Use the shown graph to solve the given simultaneous equations.

  • A π‘₯ = 1 and 𝑦 = 2
  • B π‘₯ = 2 and 𝑦 = 0
  • C π‘₯ = 2 and 𝑦 = 1
  • D π‘₯ = 0 and 𝑦 = 2
  • E π‘₯ = 2 and 𝑦 = 2

Q4:

Use the shown graph to solve the simultaneous equations

  • A π‘₯ = 3 and 𝑦 = 7
  • B π‘₯ = 0 and 𝑦 = 3
  • C π‘₯ = βˆ’ 2 and 𝑦 = 3
  • D π‘₯ = 3 and 𝑦 = 0
  • E π‘₯ = 7 and 𝑦 = 3

Q5:

Which of the following sets of simultaneous equations could be solved using the given graph?

  • A 𝑦 = βˆ’ 4 π‘₯ + 2 , 𝑦 = 5 π‘₯ βˆ’ 1
  • B 𝑦 = 2 π‘₯ βˆ’ 4 , 𝑦 = π‘₯ + 5
  • C 𝑦 = βˆ’ 4 π‘₯ + 2 , 𝑦 = 5 π‘₯ + 1
  • D 𝑦 = 2 π‘₯ βˆ’ 4 , 𝑦 = βˆ’ π‘₯ + 5
  • E 𝑦 = 2 π‘₯ + 4 , 𝑦 = βˆ’ π‘₯ + 5

Q6:

By plotting the graphs of 3 π‘₯ + 2 𝑦 = 1 0 and 6 π‘₯ + 4 𝑦 = 3 0 , determine the pair of π‘₯ - and 𝑦 -coordinates that satisfies both equations simultaneously.

  • AThe point coordinates that satisfy both equations are (5, 0).
  • BThe point coordinates that satisfy both equations are (0, 5).
  • CThe two lines are coincident, so there is an infinite number of solutions.
  • DThe two lines are parallel, so there is no solution.
  • EThe point coordinates that satisfy both equations are (5, 5).

Q7:

Using the graph, determine which of the following is a sensible estimate for the solution to the simultaneous equations

  • A π‘₯ = βˆ’ 0 . 5 , 𝑦 = 4 . 5
  • B π‘₯ = βˆ’ 0 . 5 , 𝑦 = βˆ’ 4 . 5
  • C π‘₯ = 0 . 5 , 𝑦 = βˆ’ 5 . 5
  • D π‘₯ = 0 . 5 , 𝑦 = βˆ’ 4 . 5
  • E π‘₯ = 0 . 5 , 𝑦 = 5 . 5

Q8:

The given graph shows the lines 𝑦 = 2 π‘₯ βˆ’ 1 and 𝑦 = βˆ’ 4 π‘₯ βˆ’ 7 . Determine the point whose coordinates satisfy both equations simultaneously.

  • A ( 1 , 3 )
  • B ( βˆ’ 3 , βˆ’ 1 )
  • C ( 3 , 1 )
  • D ( βˆ’ 1 , βˆ’ 3 )
  • E ( βˆ’ 2 , βˆ’ 7 )

Q9:

Plot the graphs of the equations and then solve the system.

  • A π‘₯ = 2 , 𝑦 = βˆ’ 9
  • B π‘₯ = 1 , 𝑦 = 9
  • CThere are no solutions as both equations represent parallel lines.
  • D π‘₯ = βˆ’ 2 , 𝑦 = 3
  • EThere are infinitely many solutions as both equations represent the same line.

Q10:

Which of the following graphs could be used to help solve the set of simultaneous equations

  • A
  • B
  • C
  • D
  • E

Q11:

Do the simultaneous equations plotted in the given graph have a solution? If yes, find it.

  • Ano
  • Byes,
  • Cyes, (1, 7)
  • Dyes,
  • Eyes, (7, 1)

Q12:

Using the graph, determine which of the following is a sensible estimate for the solution to the simultaneous equations

  • A π‘₯ = 2 . 9 , 𝑦 = 2 . 1
  • B π‘₯ = 2 . 7 , 𝑦 = 2 . 3
  • C π‘₯ = 2 . 2 , 𝑦 = 2 . 8
  • D π‘₯ = 2 . 6 , 𝑦 = 2 . 4
  • E π‘₯ = 2 . 8 , 𝑦 = 2 . 3

Q13:

By graphing the functions 𝑓 ( π‘₯ ) = 3 π‘₯ βˆ’ 2 and 𝑔 ( π‘₯ ) = 4 βˆ’ π‘₯ , state the point at which they intersect.

  • A ( 3 , 7 )
  • B ( 0 . 5 , 3 . 5 )
  • C ( 1 . 7 , 2 . 1 )
  • D ( 1 . 5 , 2 . 5 )

Q14:

The given graph shows the lines 𝑦 = π‘₯ + 1 and 𝑦 = βˆ’ π‘₯ + 3 . Determine the point whose coordinates satisfy both equations simultaneously.

  • A ( 0 , 3 )
  • B ( 2 , 1 )
  • C ( 1 , 0 )
  • D ( 1 , 2 )
  • E ( 1 , 3 )

Q15:

Use the shown graph to find appropriate ranges for the solutions to the simultaneous equations

  • A βˆ’ 1 < π‘₯ < βˆ’ 0 . 7 5 and 0 . 5 < 𝑦 < 0 . 7 5
  • B 0 . 5 < π‘₯ < 0 . 7 5 and βˆ’ 3 < 𝑦 < βˆ’ 1
  • C βˆ’ 0 . 7 5 < π‘₯ < βˆ’ 0 . 5 and 0 . 7 5 < 𝑦 < 1
  • D 0 . 5 < π‘₯ < 0 . 7 5 and βˆ’ 1 < 𝑦 < βˆ’ 0 . 7 5
  • E 0 . 7 5 < π‘₯ < 1 and βˆ’ 0 . 7 5 < 𝑦 < βˆ’ 0 . 5

Q16:

The given graph shows the lines 𝑦 = 3 π‘₯ βˆ’ 1 and 𝑦 = 4 π‘₯ βˆ’ 7 . Determine the point whose coordinates are solutions to both equations simultaneously.

  • A ( 2 , βˆ’ 7 )
  • B(17, 6)
  • C ( βˆ’ 7 , 2 )
  • D(6, 17)
  • E ( 0 , βˆ’ 7 )

Q17:

By plotting the graphs of 𝑦 = βˆ’ 2 π‘₯ + 1 and 𝑦 = π‘₯ + 4 , find the point that satisfies both equations simultaneously.

  • A(1, 3)
  • B(3, 7)
  • C ( βˆ’ 3 , 7 )
  • D ( βˆ’ 1 , 3 )
  • E(3, 1)

Q18:

Determine whether the simultaneous equations plotted in the given graph have a solution.

  • AThey have an infinite number of solutions.
  • BThey have a solution.
  • CThey have two solutions.
  • DThey do not have a solution.

Q19:

By plotting the graphs of 𝑦 = π‘₯ βˆ’ 1 and 𝑦 = 5 π‘₯ + 7 , find the point that satisfies both equations simultaneously.

  • A ( 1 , βˆ’ 1 )
  • B ( βˆ’ 3 , βˆ’ 2 )
  • C ( βˆ’ 1 , 1 )
  • D ( βˆ’ 2 , βˆ’ 3 )
  • E ( 1 , 7 )

Q20:

Use the shown graph to solve the given simultaneous equations.

  • A π‘₯ = 3 and 𝑦 = 3
  • B π‘₯ = 2 and 𝑦 = 1
  • C π‘₯ = 3 and 𝑦 = 0
  • D π‘₯ = 1 and 𝑦 = 2
  • E π‘₯ = 0 and 𝑦 = 3

Q21:

Plot the graphs of the simultaneous equations and then solve the system.

  • A π‘₯ = 1 , 𝑦 = 9
  • BThere are infinitely many solutions as both equations represent the same line.
  • C π‘₯ = 1 , 𝑦 = βˆ’ 2
  • DThere are no solutions as both equations represent parallel lines.
  • E π‘₯ = 0 , 𝑦 = 7

Q22:

Using the graph, determine which of the following is a sensible estimate for the solution to the simultaneous equations

  • A π‘₯ = 5 . 9 , 𝑦 = 2 . 7
  • B π‘₯ = 5 . 4 , 𝑦 = 3 . 1
  • C π‘₯ = 5 . 6 , 𝑦 = 2 . 4
  • D π‘₯ = 5 . 7 , 𝑦 = 2 . 9
  • E π‘₯ = 5 . 4 , 𝑦 = 2 . 9

Q23:

Using the given graph, determine whether there is a point whose coordinates satisfy the equations of both lines simultaneously. If yes, find its coordinates.

  • AThe intersection point is ( 3 , βˆ’ 5 ) .
  • BThe intersection point is ( 1 , βˆ’ 2 ) .
  • CThe intersection point is ( 0 , βˆ’ 2 ) .
  • DThe two lines are parallel, so there is no such point.
  • EThe intersection point is ( 0 , βˆ’ 5 ) .

Q24:

Plot the graphs of the equations and then solve the system.

  • A π‘₯ = βˆ’ 2 , 𝑦 = 1 1
  • B π‘₯ = 2 , 𝑦 = 1 1
  • C π‘₯ = 0 , 𝑦 = 7 2
  • D π‘₯ = βˆ’ 7 2 , 𝑦 = 0
  • E π‘₯ = 2 , 𝑦 = 5

Q25:

Do the simultaneous equations 𝑦 = 1 3 π‘₯ + 4 3 and 𝑦 = βˆ’ 1 2 π‘₯ + 1 1 2 plotted in the given graph have a solution? If yes, find it.

  • Ayes, ( 5 , 5 )
  • Byes, ( 3 , 5 )
  • Cno
  • Dyes, ( 5 , 3 )
  • Eyes, ( 2 , 6 )

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