Solving Contextual Problems Using Systems and Algebraic Expressions

Practice questions and answers covering algebraic phrase translation, setting up systems of linear equations from contextual word problems, and solving systems using elimination and substitution.

What is the algebraic expression for the verbal phrase: "The sum of a number and 9"?

$$x + 9$$

What is the algebraic expression for the verbal phrase: "14 decreased by a number"?

$$14 - x$$

What is the algebraic expression for the verbal phrase: "The product of x and 7"?

$$7x$$

What is the algebraic expression for the verbal phrase: "The quotient of 36 and a number"?

$$36 \div x$$

What is the algebraic expression for the verbal phrase: "8 less than a number"?

$$x - 8$$

What is the algebraic expression for the verbal phrase: "A number j increased by 28"?

$$j + 28$$

What is the algebraic expression for the verbal phrase: "15 subtracted from a number"?

$$x - 15$$

What is the algebraic expression for the verbal phrase: "A number y split into 5 equal groups"?

$$y \div 5$$

What is the algebraic expression for the phrase: "Twice a number, increased by 6"?

$$2x + 6$$

What is the algebraic expression for the phrase: "Five times the difference between a number and 3"?

$$5(x - 3)$$

What is the algebraic expression for the phrase: "Three times A minus four"?

$$3A - 4$$

What is the algebraic expression for the phrase: "The quotient of a number and 2, added to 15"?

$$(x \div 2) + 15$$

What is the algebraic expression for the phrase: "9 centimeters less than three-fourths of a length x"?

$$\frac{3}{4}x - 9$$

What is the algebraic expression for the phrase: "8 times the sum of twice a number and 5"?

$$8(2x + 5)$$

In Problem 1, DVDs cost $15.49 each locally, or $13.99 each plus $6 shipping online. What system of equations models the total cost y for x DVDs?

$y = 15.49x$ and $y = 13.99x + 6$

How many DVDs must be bought for the total cost to be the same at the local store and online store in Problem 1?

$4\text{ DVDs}$ (calculated from $1.50x = 6$)

In Problem 2, what system of equations represents two numbers where the sum of the first and three times the second is -2, and the sum of the second and two times the first is 6?

$x + 3y = -2$ and $2x + y = 6$ (where $x$ is the first number and $y$ is the second number)

What are the values of the first number (x) and second number (y) found by solving the system in Problem 2?

$x = 4$ and $y = -2$ (giving the ordered pair $(4, -2)$)

In Problem 3, a jar has 23 total coins made of nickels (x) and dimes (y) worth $1.55. What two equations represent this scenario?

$x + y = 23$ and $0.05x + 0.10y = 1.55$ (or in cents: $5x + 10y = 155$)

In Problem 3, when using elimination with $-5(x + y = 23)$ and $5x + 10y = 155$, what equation is obtained to solve for y?

$5y = 40$ (which simplifies to $y = 8$ dimes)