Question

from unit 2, lesson 12 here are clues for a puzzle involving two numbers. - seven times the first number plus six times the second number equals 31. - three times the first number minus 10 times the second number is 29. what are the two numbers? explain or show your reasoning.

Explanation:

Step1: Define variables for numbers

Let the first number be $x$, the second be $y$.

Step2: Translate clues to equations

$$7x + 6y = 31$$
$$3x - 10y = 29$$

Step3: Eliminate one variable (e.g., $x$)

Multiply first eq by 3: $21x + 18y = 93$
Multiply second eq by 7: $21x - 70y = 203$
Subtract second from first:
$$(21x+18y)-(21x-70y)=93-203$$
$$88y = -110$$

Step4: Solve for $y$

$$y = \frac{-110}{88} = -\frac{5}{4} = -1.25$$

Step5: Substitute $y$ to find $x$

Use $3x - 10y = 29$:
$$3x - 10(-1.25) = 29$$
$$3x + 12.5 = 29$$
$$3x = 29 - 12.5 = 16.5$$
$$x = \frac{16.5}{3} = 5.5 = \frac{11}{2}$$

Answer:

The first number is $\frac{11}{2}$ (or 5.5) and the second number is $-\frac{5}{4}$ (or -1.25).