Year11 MATH 2-2-3 Warm-up Questions-Linear Equations and their Graphs (Trend Analysis).

For Chapter 3, we move into the “language of trends.” In General Mathematics, linear equations aren’t just about finding $x$ ; they are about modeling how one variable (like time) affects another (like cost or distance).

A high-achiever in this section focuses on the gradient ( $m$ ) as a “rate of change” and the $y$ -intercept ( $c$ ) as a “starting condition.”

Concepts and Skills Covered:

Algebraic Rearrangement: Solving for a specific variable in a linear model.
Gradient ( $m$ ): Calculating the rate of change using $m = \frac{y_2 – y_1}{x_2 – x_1}$ .
Intercepts: Understanding the physical meaning of where a graph hits the axes.
Simultaneous Equations: Finding the point of intersection (the “break-even” point).
Linear Modeling: Constructing an equation from a word problem.

Q1. A phone plan has a monthly fixed cost of $\$30$ and a data charge of $\$5$ per gigabyte ( $g$ ). Which equation models the total monthly cost ( $C$ )?

A. C=30g+5

B. g=5C+30

C. C=5g+30

D. C=35g

Hint: Identify the ‘starting value’ (fixed) and the ‘rate of change’ (per unit).

Q2. A line passes through the points $(2, 10)$ and $(5, 25)$ . What is the gradient ( $m$ ) of this line?

A. 5

B. 0.2

C. 15

D. 3

Hint: Use the formula: $m = \frac{y_2 – y_1}{x_2 – x_1}$ .

Q3. What is the $x$ -intercept of the equation $4x + 2y = 20$ ?

A. 5

B. 10

C. 20

D. 4

Hint: The x-intercept occurs when the y value is zero.

Q4. Solve for $x$ in the equation: $3(x – 4) = 12 + x$ .

A. 6

B. 0

C. 24

D. 12

Hint: Expand the brackets first, then group all x terms on one side.

Q5. A business has a cost function $C = 10x + 500$ and a revenue function $R = 30x$ . How many units ( $x$ ) must they sell to break even?

A. 16.6 units

B. 20 units

C. 25 units

D. 50 units

Hint: Break-even occurs when Total Cost equals Total Revenue (C=R).

Q6. If a line has a gradient of $-2$ and passes through $(3, 4)$ , what is its $y$ -intercept ( $c$ )?

A. 2

B. −10

C. 10

D. −2

Hint: Plug the known x,y, and m values into the general linear equation y=mx+c.

Q7. In a trend analysis of car value, the equation is $V = -3000t + 45000$ , where $t$ is years. What does the number $-3000$ represent?

A. The annual depreciation in dollars.

B. The price of the car after one year.

C. The total number of cars sold.

D. The original price of the car.

Hint: Consider what the ‘rate of change’ means in the context of value over time.

Q8. Which of the following lines is parallel to $y = 4x + 7$ ?

A. y=−0.25x+7

B. y=0.25x+7

C. y=−4x+7

D. y=4x−12

Hint: Look for the equation with the exact same gradient (m) value.

Q9. A line of best fit for a set of data points (10,50) to (50,200) is used to predict a value at x=100. This is an example of:

A. Interpolation

B. Correlation

C. Extrapolation

D. Simultaneous solving

Hint: Is the target value ( $x=100$ ) inside or outside the given data range?

Q10. Using the substitution method for $y = 2x$ and $x + y = 9$ , what is the value of $x$ ?

A. 9

B. 3

C. 6

D. 4.5

Hint: Replace the y in the second equation with the expression ‘2x’.

****************************************************************************

第3章では、「傾向の言語」を扱います。一般数学において、線形方程式は単に $x$ を求めるだけでなく、ある変数（時間など）が別の変数（費用や距離など）にどのように影響するかをモデル化します。このセクションの優秀な生徒は、勾配（ $m$ ）を「変化率」として、また $y$ 切片（ $c$ ）を「初期条件」として重視します。

扱われる概念とスキル：

代数的再配置：線形モデルにおける特定の変数の解を求める。

勾配（ $m$ ）： $m = \frac{y_2 – y_1}{x_2 – x_1}$ を用いて変化率を計算する。

切片：グラフが軸と交わる位置の物理的な意味を理解する。

連立方程式：交点（「損益分岐点」）を求める。

線形モデリング：文章題から方程式を構築する。

******************

Right answer

Q1. C. C=5g+30 The fixed cost is the y-intercept (30) and the rate per GB is the gradient (5).

Q2. A. 5 m=(25−10)/(5−2)=15/3=5.

Q3. A. 5

To find the $x$ -intercept, set $y = 0$ . $4x + 2(0) = 20 \implies 4x = 20 \implies x = 5$ .

Q4. D. 12 Expand: $3x – 12 = 12 + x$ . Rearrange: $2x = 24$ . Solve: $x = 12$ .

Q5. C. 25 units Set $C = R$ : $10x + 500 = 30x \implies 20x = 500 \implies x = 25$

Q6. C. 10

Substitute into $y = mx + c$ : $4 = -2(3) + c \implies 4 = -6 + c \implies c = 10$ .

Q7. A. The annual depreciation in dollars. The gradient represents the rate of change; since it is negative, the value is decreasing by $\$3,000$ each year.

Q8. $y = 4x – 12$ Parallel lines must have the same gradient ( $m=4$ ).

Q9. C. Extrapolation Predicting outside the range of known $x$ -values ( $10$ to $50$ ) is called extrapolation.

Q10. B. 3

Substitute $2x$ for $y$ in the second equation: $x + 2x = 9 \implies 3x = 9 \implies x = 3$ .

Post Views: 6