Which of the following graphs demonstrates a system of linear equations with no solution? i. ii. iii. iv.

Q: Which of the following graphs demonstrates a system of linear equations with no solution? i. ii. iii. iv.

### Related Formula A system of linear equations has no solution if and only if the lines are parallel. ### Core Logic Examine each graph for parallel lines: - Graph i shows perpendicular lines (one vertical, one horizontal). - Graph ii shows two horizontal parallel lines. - Graph iii shows two vertical parallel lines. - Graph iv shows intersecting lines. ### Step 1: Conclusion Since graphs ii and iii depict parallel lines, both systems have no solution. Hence, option D is correct. ### Pattern R

Year	2023	Total
Questions	17	17

Q10 sat_math_2023_01_january_morning Interpreting Linear Functions

In a factory, the number of units manufactured in a month follows the linear function

N = 231D

, where

N

is the number of units produced in the month and

D

is the number of days elapsed in the month. Which of the following represents 231 in the above equation?

A. The number of units manufactured only in the first day
B. The number of workers working in the factory
C. The number of units manufactured each day of the month
D. The increase in the number of workers in the factory per day

Solution

Related Formula

$N = mD$ $N = mD$

where $m$ $m$ represents the rate of unit production per day.

Core Logic

Evaluate $N$ $N$ for successive values of $D$ $D$: For $D = 1$ $D = 1$, $N = 231$ $N = 231$. For $D = 2$ $D = 2$, $N = 462$ $N = 462$.

Each elapsed day adds 231 units to total production.

Step 1: Conclusion

Thus, 231 represents the number of units manufactured each day of the month (Option C).

Pattern Recognition

Linear multiplier of time variable $arrow$ $\rightarrow$ represents daily rate.

Chapter Mix

SAT Math: Heart of Algebra

Q12 sat_math_2023_01_january_morning Solving Linear Equations

If

5a - 7 = 2a - 22

, what is the value of

a + 2

?

A. 4
B. -3
C. 0
D. 2

Solution

Related Formula

ax + b = cx + d (a - c)x = d - b

Core Logic

Solve $5a - 7 = 2a - 22$ $5a - 7 = 2a - 22$ for $a$ $a$:

5a - 2a = -22 + 7

3a = -15 a = -5

Step 1: Calculate Target Expression

Substitute $a = -5$ $a = -5$ into $a + 2$ $a + 2$:

a + 2 = -5 + 2 = -3

Thus, option B is correct.

Pattern Recognition

Trap alert: do not stop at $a = -5$ $a = -5$; always check what the question asks for ( $a + 2$ $a + 2$).

Chapter Mix

SAT Math: Heart of Algebra

Q13 sat_math_2023_01_january_morning Interpreting Linear Functions

P = 89 + 12t

The number of vehicles in public transportation in City X between 2005 and 2014 is modeled by the equation above. In the equation,

P

is the number of vehicles and

t

is the year (assuming 2005 is

t = 0

). What does the 12 mean in the equation?

A. The number of vehicles in public transportation was 12 in 2005.
B. The number of vehicles in public transportation increased by 12 vehicles each year between 2005 and 2014.
C. The number of vehicles in public transportation increased in total by 12 between 2005 and 2014.
D. The number of vehicles in public transportation was 12 in 2014.

Solution

Related Formula

$y = mt + b$ $y = mt + b$

where $m$ $m$ represents the constant rate of change per unit of time $t$ $t$.

Core Logic

In $P = 89 + 12t$ $P = 89 + 12t$, $89$ $89$ is the initial value at $t = 0$ $t = 0$ (year 2005), and $12$ $12$ is the rate of change per year.

Each increase of $1$ $1$ in $t$ $t$ causes $P$ $P$ to increase by $12$ $12$.

Step 1: Conclusion

Thus, 12 means the number of vehicles increased by 12 vehicles each year between 2005 and 2014 (Option B).

Pattern Recognition

Coefficient of time variable $t$ $t$ $arrow$ $\rightarrow$ represents annual rate of increase/decrease.

Chapter Mix

SAT Math: Heart of Algebra

Q14 sat_math_2023_01_january_morning Functions

f(x) = x² + px + q

If

p = 2q

and the function has a value of 16 at

x = 3

, what is the value of

p

?

Numerical Answer. Answer: 2 to 2

Solution

Related Formula

f(3) = (3)² + p(3) + q = 16

Given substitution relationship $p = 2q$ $p = 2q$.

Core Logic

Substitute $p = 2q$ $p = 2q$ into $f(x)$ $f(x)$:

f(x) = x² + 2qx + q

Substitute $x = 3$ $x = 3$ and $f(3) = 16$ $f(3) = 16$:

16 = 3² + 2q(3) + q

$16 = 9 + 6q + q$ $16 = 9 + 6q + q$

$16 = 9 + 7q$ $16 = 9 + 7q$

7q = 7 q = 1

Step 1: Calculate p

Calculate $p$ $p$ using $p = 2q$ $p = 2q$:

$p = 2(1) = 2$ $p = 2(1) = 2$

Thus, the answer is 2.

Pattern Recognition

Substitute known relation $p = 2q$ $p = 2q$ to convert function into single parameter equation.

Chapter Mix

SAT Math: Heart of Algebra

Q15 sat_math_2023_01_january_morning Systems of Linear Equations

In the TAS aptitude test, three marks are awarded for a correct response and one mark is deducted for an incorrect response. John took the test and secured 76 marks. If there are 40 questions in the test and John attempted all questions, how many of his responses were incorrect?

Numerical Answer. Answer: 11 to 11

Solution

Related Formula

$c + i = N$ $c + i = N$ $3c - 1i = S$ $3c - 1i = S$

where $c$ $c$ is correct responses, $i$ $i$ is incorrect responses, $N$ $N$ is total questions, and $S$ $S$ is score.

Core Logic

Set up system of equations:

c + i = 40 (1)

3c - i = 76 (2)

Add (1) and (2) to eliminate $i$ $i$:

4c = 116 c = 29

Step 1: Solve for Incorrect Responses

Substitute $c = 29$ $c = 29$ into equation (1):

29 + i = 40 i = 40 - 29 = 11

John had 11 incorrect responses.

Pattern Recognition

Test marking system $arrow$ $\rightarrow$ correct + incorrect = total questions; 3(correct) - 1(incorrect) = score.

Chapter Mix

SAT Math: Heart of Algebra

Solution & Explanation

Related Formula

Core Logic

Step 1: Conclusion

Pattern Recognition

Chapter Mix

More Heart of Algebra Previous-Year Questions — Page 3

Solution

Related Formula

Core Logic

Step 1: Conclusion

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Calculate Target Expression

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Conclusion

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Calculate p

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Solve for Incorrect Responses

Pattern Recognition

Chapter Mix

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