Simultaneous Linear Equations

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Simultaneous Linear Equations — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionChecking Solutions for Both EquationsConcept Practice

2 marks~3 minCriterion A

A city planner models two proposed road alignments on a coordinate grid. Road 1 follows

y = 2x + 1

and Road 2 follows

y = -x + 4

. A junction is planned at point

P(1, 3)

.

Justify that

P(1, 3)

is the junction point of the two roads by verifying it satisfies both equations. [2]

Solutions

2QuestionSolving GraphicallyAssessment Practice

4 marks~6 minCriterion C

A city planner models two proposed road alignments as straight lines on a coordinate grid. The graph shows both lines.

Line 1 passes through

(0, 1)

and

(2, 5)

.
Line 2 passes through

(0, -3)

and

(2, 1)

Deduce the gradient of each line from the graph. [1]

Explain why the two roads will never intersect, using the properties of both lines. [2]

Advise the city planner whether a single crossing point between the two roads can ever be achieved by extending them indefinitely, and justify your advice. [1]

Solutions

3QuestionChecking Solutions for Both EquationsAssessment Practice

4 marks~6 minCriterion D

A city planner models two road-widening proposals. Proposal A follows the cost equation

y = x - 1

and Proposal B follows

y = -x + 5

, where

x

is the number of months into the project and

y

is the projected budget surplus (in millions of dollars). The graph shows both lines intersecting at point

P

State the coordinates of

P

from the graph. [1]

Show that

P

satisfies both equations simultaneously. [2]

Interpret what the coordinates of

P

mean in this context, and advise the city planner whether

P

should be used as the basis for deciding when both proposals are financially equivalent. [1]

Solutions

4QuestionSolving by Substitution MethodAssessment Practice

4 marks~6 minCriterion B

A logistics company tracks two delivery routes. Each day, the total distance covered by both routes is

a

km, and the difference between the longer and shorter route is

b

km. The distances

x

(longer) and

y

(shorter), in km, satisfy:

x + y = a \qquad x - y = b

Deduce the values of

x

and

y

for each system. [1]

System 1:

a = 5,\ b = 1

\quad

System 2:

a = 7,\ b = 3

System 3:

a = 10,\ b = 4

\quad

System 4:

a = 12,\ b = 2

Deduce general formulas for

x

and

y

in terms of

a

and

b

. [1]

Prove that your formulas satisfy both original equations. [1]

The company requires the shorter route to be at least 30% of the total daily distance. For

a = 40

and

b = 10

, advise the company whether this scheduling requirement is met, justifying your answer using your formulas. [1]

Solutions

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