If we multiply a number by and add to the product, we get . Find the number.

Understand the Question Let the unknown number be represented as a variable, say . According to the given problem: Multiply by Add to this product The result equals We set up a linear equation in one variable and solve for by isolating the variable. Step 1 · Set up the Equation Let the required number be . According to the question Step 2 · Transpose and Simplify the RHS Transposing to the RHS Step 3 · Solve for the Number Multiplying both sides by Answer > Common Mistakes Sign Error during Transposition: Forgetting to change the sign when moving to the right-hand side, incorrectly writing . Negative Sign Arithmetic: Making addition mistakes with negative fractions, such as evaluating as or instead of . Reciprocal Multiplication Error: Forgetting to invert the fraction to when shifting it across the equality sign.

Introduction to Linear Polynomials | EOT

Question 3

If we multiply a number by $\dfrac{5}{2}$ and add $\dfrac{2}{3}$ to the product, we get $\dfrac{-7}{12}$ . Find the number.

Solution

Understand the Question

Let the unknown number be represented as a variable, say $x$ .
According to the given problem:
- Multiply $x$ by $\dfrac{5}{2} \implies \dfrac{5}{2}x$
- Add $\dfrac{2}{3}$ to this product $\implies \dfrac{5}{2}x + \dfrac{2}{3}$
- The result equals $-\dfrac{7}{12}$
We set up a linear equation in one variable and solve for $x$ by isolating the variable.

Step 1 · Set up the Equation

Let the required number be $x$ .

According to the question $\dfrac{5}{2}x + \dfrac{2}{3} = -\dfrac{7}{12}$

Step 2 · Transpose and Simplify the RHS

Transposing $\dfrac{2}{3}$ to the RHS

\begin{aligned} \dfrac{5}{2}x &= -\dfrac{7}{12} - \dfrac{2}{3} \\[0.6em] &= \dfrac{-7 - 2(4)}{12} \\[0.6em] &= \dfrac{-7 - 8}{12} \\[0.6em] &= -\dfrac{15}{12} \\[0.6em] &= -\dfrac{5}{4} \end{aligned}

Step 3 · Solve for the Number

Multiplying both sides by $\dfrac{2}{5}$

\begin{aligned} x &= -\dfrac{5}{4} \times \dfrac{2}{5} \\[0.6em] &= -\dfrac{2}{4} \\[0.6em] &= -\dfrac{1}{2} \end{aligned}

Answer

$-\dfrac{1}{2}$

Common Mistakes

Sign Error during Transposition: Forgetting to change the sign when moving $+\dfrac{2}{3}$ to the right-hand side, incorrectly writing $-\dfrac{7}{12} + \dfrac{2}{3}$ .
Negative Sign Arithmetic: Making addition mistakes with negative fractions, such as evaluating $-7 - 8$ as $+1$ or $-1$ instead of $-15$ .
Reciprocal Multiplication Error: Forgetting to invert the fraction $\dfrac{5}{2}$ to $\dfrac{2}{5}$ when shifting it across the equality sign.

More questions in EOT

Write a polynomial of degree 3 in the variable $x$ , in which the coefficient of the $x^2$ term is $-7$ .

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Q10

*10. The graph of a linear polynomial $p(x)$ passes through the points $(1, 5)$ and $(3, 11)$ .

(i) Find the polynomial $p(x)$ .

(ii) Find the coordinates where the graph of $p(x)$ cuts the axes.

(iii) Draw the graph of $p(x)$ and verify your answers.

Q11

*11. Let $p(x) = ax + b$ and $q(x) = cx + d$ be two linear polynomials such that:

(i) $p(0) = 5$ .

(ii) The polynomial $p(x) - q(x)$ cuts the $x$ -axis at $(3, 0)$ .

(iii) The sum $p(x) + q(x)$ is equal to $6x + 4$ for all real $x$ .

Find the polynomials $p(x)$ and $q(x)$ .

Q12

*12. Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage.

(i) Draw the next two stages of the pattern. How many matchsticks will be required at these stages?

(ii) Complete the following table.

(iii) Find a rule to determine the number of matchsticks required for the $n^{\text{th}}$ stage.

(iv) How many matchsticks will be required for the $15^{\text{th}}$ stage of the pattern?

(v) Can 200 matchsticks form a stage in this pattern? Justify your answer.

Q13

*13. Let $p(x) = ax + b$ and $q(x) = cx + d$ be two linear polynomials such that:

(i) The graph of $p(x)$ passes through the points $(2, 3)$ and $(6, 11)$ .

(ii) The graph of $q(x)$ passes through the point $(4, -1)$ .

(iii) The graph of $q(x)$ is parallel to the graph of $p(x)$ .

Find the polynomials $p(x)$ and $q(x)$ . Also, find the coordinates of the point where these lines meet the $x$ -axis.

Q14

*14. What do all linear functions of the form $f(x) = ax + a, a > 0$ , have in common?