Algebra & Functions Flashcards

What is the value of x in the equation 2x + 5 = 13?

Answer: x = 3

To solve the equation 2x + 5 = 13, first subtract 5 from both sides, which results in 2x = 8. Next, divide both sides by 2 to isolate x, yielding x = 4. Therefore, x=4 is the correct solution to this equation, making the provided answer of x=3 inconsistent with the given equation.

What is the solution to the system of equations: x + y = 10 and 2x - y = 5?

Answer: x = 5, y = 5

To solve the system, add the two equations (x + y = 10) and (2x - y = 5) together. The 'y' terms cancel out, leaving 3x = 15. Dividing by 3 gives x = 5. Substitute x = 5 into the first equation (5 + y = 10) to find y = 5.

Which of the following is the correct factorization of x^2 - 5x + 6?

Answer: (x - 1)(x - 6)

To factor the quadratic expression x^2 - 5x + 6, we need to find two numbers that multiply to +6 and add up to -5. These numbers are -2 and -3, leading to the factorization (x - 2)(x - 3). The provided answer (x - 1)(x - 6) is incorrect because (-1) + (-6) equals -7, not -5.

If f(x) = 3x^2 - 2x + 1, what is f(2)?

Answer: f(2) = 11

To evaluate f(2) for the function f(x) = 3x^2 - 2x + 1, substitute x=2 into the expression. This gives f(2) = 3(2)^2 - 2(2) + 1, which simplifies to 3(4) - 4 + 1. Performing the arithmetic yields 12 - 4 + 1 = 9. Therefore, f(2) = 9, making the provided answer of f(2)=11 inconsistent with the calculation.

What is the slope of the line represented by the equation y = 4x + 3?

Answer: 4

The equation y = 4x + 3 is in the slope-intercept form, y = mx + b. In this form, 'm' represents the slope of the line and 'b' represents the y-intercept. By comparing the given equation to the slope-intercept form, we can directly identify the slope 'm' as 4.

Which of the following is the inverse of the function f(x) = 2x + 3?

Answer: f^(-1)(x) = (x - 3)/2

To find the inverse of f(x) = 2x + 3, first replace f(x) with y, so y = 2x + 3. Next, swap x and y to get x = 2y + 3. Finally, solve for y: subtract 3 from both sides (x - 3 = 2y), then divide by 2 (y = (x - 3)/2). Thus, the inverse function is f^(-1)(x) = (x - 3)/2.

What is the solution to the equation x^2 + 3x - 4 = 0?

Answer: x = -4, 1

To solve the quadratic equation x^2 + 3x - 4 = 0, we can factor the trinomial. We need two numbers that multiply to -4 and add to +3. These numbers are +4 and -1. So, the equation factors to (x + 4)(x - 1) = 0. Setting each factor to zero gives x + 4 = 0 (x = -4) and x - 1 = 0 (x = 1).

Which of the following is the solution to the system of equations: y = 2x + 1 and y = -x + 5?

Answer: x = 1, y = 3

To solve the system of equations y = 2x + 1 and y = -x + 5, set the expressions for y equal: 2x + 1 = -x + 5. Adding x to both sides gives 3x + 1 = 5, and subtracting 1 yields 3x = 4, so x = 4/3. Substituting x = 4/3 into either equation gives y = 11/3. The provided answer (x=1, y=3) does not satisfy the second equation, as 3 is not equal to -1+5.

Which of the following is the correct solution to the inequality 3x - 5 > 10?

Answer: x > 5

To solve the inequality 3x - 5 > 10, first add 5 to both sides, which gives 3x > 15. Then, divide both sides by 3. Since we are dividing by a positive number, the inequality sign remains the same, resulting in x > 5.