Download Grade 9 Linear Equations in Two Variables

ID : ph-9-Linear-Equations-in-Two-Variables [1]
Grade 9
Linear Equations in Two Variables
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Answer t he quest ions
(1)
Find the linear equation represented in the graph below
(2)
At what point does line represented by the equation 7x + 7y = 56 intersects a line which is
parallel to the x-axis, and at a distance 1 units f rom the origin and in the negative direction of yaxis.
(3)
If point (3, 5) lies on the graph of linear equation 2x + b y = 16, f ind the value of b .
(4) In the graph of the linear equation 5x + 4y = 38, there is a point such that its ordinate is 5 more
than its abscissa. Find coordinates of that point.
Choose correct answer(s) f rom given choice
(5)
If graph of the equation y = mx + c passes through the origin, what is the value of c.
a. 2
b. 0
c. -1
d. 1
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ID : ph-9-Linear-Equations-in-Two-Variables [2]
(6) If 3x + 2y = 9 then which of the f ollowing x and y values is true?
a. x=3, y=
3
b. x=
2
c. x=2, y=
3
3
, y=
3
2
d. x=5, y=
2
4
4
2
(7) A telecom operator charges ₱ 1.3 f or the f irst minute and ₱ 0.8 per minute f or subsequent
minutes of a call. If duration of call is represented as d, and amount charged is represented as
c, f ind the linear equation f or this relationship.
(8)
a. c = 1.3d + 0.8
b. c = 0.8d + 0.5
c. c = 0.8d + 1.3
d. c = 1.3d + 0.5
If a number is added to both side of a equation, then solution of the equation
a. Remains the same
b. Changes
c. May or may not change depending on the
equation
d. Will also increase by same number
(9) T he equation of x-axis is
a. x = y
b. x = 0
c. y = 0
d. x + y = 0
(10) A point of the f orm (0, p) lies on the line
a. x = 0
b. x + y = 0
c. y = 0
d. x = y
(11) Find the point where linear equation 3x + 3y = 18 intersects with y-axis.
a. (0, 3)
b. (6, 0)
c. (0, 6)
d. (3, 0)
(12) In graph of linear equation 3x + 5y = 90, there is a point such that its ordinate is thrice of
abscissa. Find coordinates of the point.
a. (15, 5)
b. (12, 4)
c. (4, 12)
d. (5, 15)
(13) Equation 4x + 3y = 7 has a unique solution if x and y are
a. Positive Real Numbers
b. Real Numbers
c. Natural Numbers
d. Rational Numbers
(14) T he positive solutions of the equation px + qy + r = 0 always lie in
a. T hird quadrant
b. Second quadrant
c. Fourth quadrant
d. First quadrant
(15) A line passe through points (-1, -2) and (-3, 0). Find the x-intercept of the line.
a. -2.5
b. -2
c. -3
d. -4
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ID : ph-9-Linear-Equations-in-Two-Variables [3]
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generated at www.edugain.com
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ID : ph-9-Linear-Equations-in-Two-Variables [4]
Answers
(1)
y=1
T he general equation of a line is y=mx+c
So we have to f ind m and c
T o f ind c, note f rom the equation that c is the value of y when x=0 (i.e. the equation
becomes y=m*0 + c, or y=c).
Look at the graph to see if this is a vertical line. If it is not (we'll see the case where it is later
in this tip), then what the value of y is when the equation crosses the vertical axis
We see that the value of y at this point is 1. So c=1
T he next part is f inding m
T he best way to consider m is to think of it as the slope of the line.
T hink of it as the change in y f or a given change in x.
Consider the two equations,
y1 = mx1 + c, and
y2 = mx2 + c
Now we subtract the f irst equation f rom the second
We get y1 - y2 = mx1 + c - (mx2 + c)
Simplif ying,
(y1 - y2) = m(x1 - x2)
or m = (y1 - y2)/(x1 - x2)
Now, substitute the two points seen in the graph.
m = (1 - (1))/(1 - (-1))
Also, note that this is the reason why we don't apply this when the line is vertical, because
the denominator would be 0, and the equation is meaningless
T his is solved to get the value of m, and get the answer m=0
Now, if the line is a vertical one, then you can solve it by inspection.
So the answer is y=1.
(2)
(9, -1)
Let's consider the second line f irst.
T he line which is parallel to the x-axis and is at a distance 1 units f rom the origin in the
negative direction of the y-axis is def ined by the f ollowing equation
y=-1 So, now we know that at the point of intersection, the value of y = -1
T he equation of the f irst line is
7x + 7y = 56
Subtituting f or y with the value -1 in this equation, we get
x=9
So the answer is that the intersection is at the point (9, -1)
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ID : ph-9-Linear-Equations-in-Two-Variables [5]
(3)
2
We know the f ollowing f acts
- T he equation of the line is 2x + b y = 16
- T he point (3,5) lies on the line
Substitute x=3 and y=5 in the equation
2 x 3 + b x 5 = 16
Solve this to f ind that the value of b is 2.
(4) (2, 7)
Step 1
We are given the f ollowing f acts:
T he equation is 5x + 4y = 38
T he line has a point where the value of the ordinate is 5 more the value of the abscissa
T he second f act implies the point is of the f orm (x,x + 5)
Step 2
Substituting this into the equation, we get
5x + 4(x - 5) = 38
Step 3
Solving f or this gets us the value of x = 2.
From this we can f ind y = x + 5 = 7
(5)
b. 0
Step 1
For a line to pass through point (0,0), it's equation need to satisf y f or x = 0 and y = 0
Step 2
Lets substitute these values of x and y in the equation and check,
y = mx + c
⇒ 0 = m×0 + c
⇒c=0
Step 3
T heref ore, the graph of the equation y = mx + c will pass through the origin if value of c is
0
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ID : ph-9-Linear-Equations-in-Two-Variables [6]
(6)
c. x=2, y=
3
2
Step 1
Remember that f or equations in two variables, you need two equations to get the unique
answer. Here we are given just one equation, so this inf ormation is not enough f or us to
solve f or a unique answer. However, we are given f our choices here, so we can quickly
substitute the f our options to see which one satisf ies the equation
Step 2
Among given f our option we can see that only x=2, y=
3
satisf ies given equation,
2
3x + 2y = 9
⇒3×2+2×
3
=9
2
⇒9=9
(7) b. c = 0.8d + 0.5
Step 1
We are given the f ollowing f acts
- T he charge f or the f irst minute is is ₱ 1.3
- T he charge per minute af ter that is ₱ 0.8
Step 2
We can see that the charge will be dependent on the time spent in minutes
So we set c on the lef t hand side
c = Some linear f unction of d
Step 3
We know that af ter the f irst minute, the rate is ₱ 0.8 per minute.
So if the call lasts f or d minutes, there will be a charge ₱ 1.3 f or the f irst minute, and ₱ 0.8
f or d - 1 minutes.
Step 4
T his means the equation is c = 1.3 + ((d - 1) x 0.8)
Step 5
Simplif ying, we get c = 0.8d + 0.5
(8)
a. Remains the same
T hink of this in simple terms.
If two values (let it be anything - weights, lengths, coins, equations etc.) are equal, and you
add or remove some amount f rom both of them, the resulting values will also be equal.
T hat is the principle here, and the answer is that the solution to the equation will remain the
same.
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ID : ph-9-Linear-Equations-in-Two-Variables [7]
(9) c. y = 0
T ake a look at a graph
You can see that f or the x-axis, the value of y is always 0. So the equation is y=0
(10) a. x = 0
T here are of course, inf inite lines that can pass through a given point, but we have to
choose f rom the f our possibilities presented. T he point specif ied is ((0, p)). Out of the f our
options the only one it actually can match is x = 0
(11) c. (0, 6)
We are told to f ind the point where the equation intersects with the y axis.
Now, at that point the value of x will be zero.
So we need to substitute x=0 into the equation.
From there, we can then solve to f ind the value of y to be 0. So the point is (0,6)
(12) d. (5, 15)
We are given the f ollowing f acts:
T he equation is 3x + 5y = 90
T he line has a point where the value of the ordinate is thrice the value of the abscissa
T he second f act implies the point is of the f orm (x,3x)
Substituting this into the equation, we get
3x + 15x = 90
Solving f or this gets us the value of x = 5. From this we can f ind y = thrice x = 15
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ID : ph-9-Linear-Equations-in-Two-Variables [8]
(13) c. Natural Numbers
A general equation in two variables has inf initely many solutions if there is no restriction
placed on the values of the two variables (x and y here). However, it may have a unique
solution if certain constraints are placed on it. Here we can see by observation that if x and
y are constrained to be natural numbers, then it has a solution f or x=y=1, and this is the
only possible solution f or natural numbers.
(14) d. First quadrant
(15) c. -3
Step 1
Equation of line y = m x + c
Step 2
Substitute f irst point in the equation
-2 = -1 m + c
m = (-2 - c)/-1 ________________(1)
Step 3
Substitute second point in the equation
0 = -3 m + c
m = (0 - c)/-3 ________________(2)
Step 4
On equating value of m f rom both equations,
(-2 - c)/-1 = (0 - c)/-3
6 + 3c = 0 + 1c
2 c = -6
c = -3
Step 5
m = (-2 - c)/-1 = -1
Step 6
Equation of line : y = -1 x + c
Now when line intersect with x axis, value of y will be 0
0 = -1x + (-3)
x = -3
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