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1. True or False:
The ordered pair (-4, 5) is a solution of the following system of
equations:
3X - 4Y= -32
-7X + 10Y = 78
We can substitute x and y values to the equation to verify if the pair
is solution or not.
We know x = -4 and y = 5, then to the first equation
3 * (-4) – 4 * 5 = -32
-7 * (-4) + 10 *5 = 28 + 50 = 78
Therefore, the pair (-4, 5) satisfies both equations. It is a solution
of the system.
2. Which one is addition method and which one is substitution
method?
(20x -3y = 14) and (4x + 10y = -8)
or
(x = 3y + 14) and (4x + 10y = -8)
The second one (x = 3y + 14) and (4x + 10y = -8)
is substitution method. In the first equation, x is expressed as a
function of y, we then can substitute it for x in the second equation
to solve y.
(20x -3y = 14) and (4x + 10y = -8)
The above system is addition method.
We can multiply -5 to the second equation, then add it to the first
equation to cancel out x terms and solve for y.
3. What is the slope of the line passing A(5,0), and B(9,40).
y yB  y A 40  0 40



 10
The slope is m 
x xB  x A
95
4
So the slope of the line passing through the above two points is 10.
4. Which one is: Parallel, Perpendicular, or neither.
x = -3y + 4
y = -3x -3
y= 3x + 2
When two lines are parallel, their slopes are equal.
When two lines are perpendicular, the product of their slopes is
equal to -1.
We first write all line equations in the slope-intercept form:
1
3
x  3 y  4  3 y   x  4  y   x 
3
4
So the slope is -1/3
The slope of line y = -3x – 3 is -3
The slope of the line y = 3x + 2 is 3
We know that (-1/3) * 3 = -1
Therefore, the lines x = -3y + 4 and y = 3x + 2 are perpendicular.
5. Solve the system by any method.
x= 7/4 y + 3
4x - 7y =12
We can solve it by substitution method because the first equation is
in the form of x as polynomial of y.
Substituting it to the second equation
7

4  y  3   7 y  12
4


7 y  12  7 y  12
00
The above equation is always true. So the system has infinite
numbers of solutions given 4x – 7y = 12. It is call underdetermined system.
6. What is the slope of the line passing through the points A(5,2)
and B(0,0).
y y A  yB 2  0 2



The slope is m 
x x A  xB 5  0 5
7. Determine whether the graphs of the pair of equations are
parallel, perpendicular or neither.
9x + 3y = 9
3x - 9y = 1
Write the equations in the slope-intercept form y = mx + b.
9 x  3 y  9  3 y  9 x  9  y  3x  1 , so the slope m1 = -3.
3x  9 y  1  9 y  3x  1  y 
1
1
x  , sol the slope is m2 = 1/3.
3
9
1
m1m2  3*  1
3
Therefore, the two lines are perpendicular.
8. Write an equation of the line that passes through the given point
and is parallel to the given line. Write the answer in slope-intercept
form.
(0,6), 4y - 12x = 2
4 y  12 x  2  4 y  12 x  2  y  3 x 
1
2
So the slope of the line 4y – 12x = 2 is 3.
The unknown line is parallel to the above line, then its slope is 3 too.
The line passes through (0, 6), so its y-intercept b = 6.
In the slope-intercept form y = mx + b
m = 3, b = 6.
So the line equation is y = 3x + 6.
9. If the following system is to be solved using the addition method,
by what constant should one of the equations be multiplied if the x
terms are to drop out?
-6x - 2y= -8
-12x - 3y= 3
The coefficient the x in the first equation is -6. The coefficient of x in the second
equation is -12. so we can multiply -2 to the first equation to obtain
-6x * (-2) – 2y * (-2) = -8 *(-2)
12x + 4y = 16
Adding the above equation to the second equation
4y – 3y = 19
y = 19
Substituting y = 19 to the first equation
-6x – 2 * 19 = -8
-6x = 30
x = -5
so the solution is x = -5, y = 19