Create linear, quadratic, and exponential equations and inequalities from a verbal context, solve them, and interpret the solution back in the situation with units.

What this topic is asking

The A-CED standards (Creating Equations) ask you to build an equation or inequality from a real-world situation, solve it, and interpret the answer in context. On the Grade 10 MCAS this modeling skill is tested in selected-response items that ask you to match a context to an equation, and in constructed-response problems scored with a rubric that rewards defining variables, writing the model, and stating the answer with meaning and units.

Translating words into a model

The first job is to recognize which kind of relationship the words describe.

• Linear: a fixed amount plus a per-unit rate. "A fee of \30 plus \0.10 per minute" is $C = 30 + 0.10m$. The fixed part is the constant; the rate multiplies the variable.
• Quadratic: an area, a product of two varying lengths, or a projectile height. "Length 3 more than width, area 70" gives $w(w + 3) = 70$.
• Exponential: a quantity that multiplies by a fixed factor each period. "A population of 500 growing 6% per year" is $P = 500(1.06)^t$.

Keywords help, but reading for the structure is more reliable: ask what stays fixed, what is added per unit, and what is multiplied repeatedly.

Defining variables and writing the model

A strong constructed-response answer always begins with a let statement: "Let $w$ = the width of the garden in feet." This names the unknown and fixes the units, and the MCAS rubric awards credit for it. Then write the equation or inequality that captures every condition in the problem.

Interpreting and discarding solutions

The final step turns a number into a meaningful answer. State it with units and check it against the situation. In a quadratic context, reject any root that has no physical meaning: a negative width, a negative time before launch, or a count that is not a whole number.

For the garden problem $w^2 + 3w - 70 = 0$, the roots are $w = 7$ and $w = -10$. A width cannot be negative, so the width is 7 feet and the length is 10 feet. Leaving the negative root in the final answer, or not stating units, is where the interpretation credit is lost.

Setting up systems from context

Some contexts need two equations. "240 tickets sold for \1500, adults \8 and students \$5" becomes the system$a + s = 240$and$8a + 5s = 1500$, with$a$and$s$ defined. The same discipline applies: define both variables, write one equation per condition, solve, and interpret.

Try this

Q1. A phone plan is \20 plus \0.05 per text. Write the cost $C$ for $t$ texts.

• Cue. $C = 20 + 0.05t$.

Q2. A square has area 81 square centimeters. Write and solve an equation for its side $s$.

• Cue. $s^2 = 81$, so $s = 9$ cm (reject $s = -9$).