Flashcard Set

SAT Math: Heart of Algebra

14 cards

Exam-grade practice on the digital SAT's Heart of Algebra domain — linear equations, inequalities, systems, and linear functions.

1
Term
In the context of a real-world linear model y = mx + b, what physical or initial quantity does the y-intercept (b) represent?
Definition
The initial or starting value of y when x = 0.
2
Term
In the context of a real-world linear model y = mx + b, what does the slope (m) represent?
Definition
The rate of change of y per unit increase in x.
3
Term
If a system of two linear equations has no solution, their graphs on the xy-plane must be ___.
Definition
parallel lines (meaning they have the exact same slope but different y-intercepts)
4
Term
The system of equations below has infinitely many solutions. What is the value of a?

3x - y = 6 ax - 2y = 12
Definition
a = 6

Reasoning: For a system to have infinitely many solutions, the equations must be equivalent. Multiplying the first equation by 2 yields 6x - 2y = 12. Comparing this to the second equation, a must equal 6.
5
Term
A line passes through the points (0, 4) and (3, 10). What is the equation of this line in slope-intercept form?
Definition
y = 2x + 4

Reasoning:

Find the slope (m): (10 - 4) / (3 - 0) = 6 / 3 = 2.

Identify the y-intercept (b): the point (0, 4) gives b = 4.
6
Term
For the linear equation 2x + 5y = 10, what are the coordinates of the x-intercept and the y-intercept?
Definition
x-intercept: (5, 0) y-intercept: (0, 2)

Reasoning:

Find the x-intercept by setting y = 0: 2x = 10 -> x = 5.

Find the y-intercept by setting x = 0: 5y = 10 -> y = 2.
7
Term
If the equation a(x + 3) = 5x - 8 has no solution for x, what is the value of the constant a?
Definition
a = 5

Reasoning: Distribute to get ax + 3a = 5x - 8. For a linear equation in one variable to have no solution, the coefficients of x on both sides must be equal (a = 5) while the constant terms must be unequal (3a != -8, which holds true as 15 != -8).
8
Term
To find the slope of a line that is perpendicular to a line with a non-zero slope m, you must find the ___.
Definition
negative reciprocal (or -1/m)
9
Term
Line L has the equation y = 3x - 7. Line M is perpendicular to Line L and passes through the point (6, 2). What is the equation of Line M in slope-intercept form?
Definition
y = -1/3x + 4

Reasoning:

The slope of M is the negative reciprocal of 3, which is -1/3.

Use point-slope form: y - 2 = -1/3(x - 6).

Simplify: y - 2 = -1/3x + 2 -> y = -1/3x + 4.
10
Term
A system of inequalities consists of y > 2x + 1 and y < -x + 4. Is the point (1, 2) a solution to this system?
Definition
No

Reasoning: Substitute (1, 2) into both inequalities:

2 > 2(1) + 1 -> 2 > 3 (False). Since the point does not satisfy the first inequality, it is not a solution to the system.
11
Term
Translate this word problem into a system of equations: 'A theater sold a total of 150 tickets. Adult tickets (A) cost $12 each and child tickets (C) cost $8 each. The total revenue was $1,400.'
Definition
A + C = 150 12A + 8C = 1400

Reasoning:

Ticket quantity equation: The sum of adult and child tickets is 150.

Total revenue equation: The revenue from adult tickets (12A) plus child tickets (8C) is 1,400.
12
Term
The system of equations below has no solution. What is the value of k?

kx - 3y = 4 4x - 6y = 9
Definition
k = 2

Reasoning: For a system of two linear equations to have no solution, the coefficients of the variables must be proportional while the constants are not. Multiplying the first equation by 2 gives: 2kx - 6y = 8. Comparing this to 4x - 6y = 9, the y-coefficients match, so we set 2k = 4, yielding k = 2 (and the constants 8 and 9 do not match, confirming no solution).
13
Term
The graph of the inequality y >= 2x - 5 includes the boundary line and all points located ___ the line.
Definition
above (or on and above)
14
Term
If f(x) = 4x - 3, how does the graph of g(x) = f(x + 2) differ from the graph of f(x) on the xy-plane?
Definition
The graph of g(x) is shifted 2 units to the left.

Reasoning: In function notation, replacing the input variable x with (x + h) results in a horizontal shift. When h is positive (+2), the shift is to the left by h units.