{"id":972,"date":"2026-01-04T14:40:43","date_gmt":"2026-01-04T04:40:43","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=972"},"modified":"2026-01-04T14:40:43","modified_gmt":"2026-01-04T04:40:43","slug":"year11-math-2-2-3-warm-up-questions-linear-equations-and-their-graphs-trend-analysis","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=972","title":{"rendered":"Year11 MATH 2-2-3 Warm-up Questions-Linear Equations and their Graphs (Trend Analysis)."},"content":{"rendered":"\n

For Chapter 3, we move into the “language of trends.” In General Mathematics, linear equations aren’t just about finding x<\/mi>x<\/annotation><\/semantics><\/math>; they are about modeling how one variable (like time) affects another (like cost or distance).<\/p>\n\n\n\n

A high-achiever in this section focuses on the gradient (<\/strong>m<\/mi>m<\/annotation><\/semantics><\/math>)<\/strong> as a “rate of change” and the y<\/mi>y<\/annotation><\/semantics><\/math>-intercept (<\/strong>c<\/mi>c<\/annotation><\/semantics><\/math>)<\/strong> as a “starting condition.”<\/p>\n\n\n\n

Concepts and Skills Covered:<\/strong><\/h3>\n\n\n\n
    \n
  1. Algebraic Rearrangement:<\/strong> Solving for a specific variable in a linear model.<\/li>\n\n\n\n
  2. Gradient (<\/strong>m<\/mi>m<\/annotation><\/semantics><\/math>):<\/strong> Calculating the rate of change using m<\/mi>=<\/mo>y<\/mi>2<\/mn><\/msub>\u2212<\/mo>y<\/mi>1<\/mn><\/msub><\/mrow>x<\/mi>2<\/mn><\/msub>\u2212<\/mo>x<\/mi>1<\/mn><\/msub><\/mrow><\/mfrac><\/mrow>m = \\frac{y_2 – y_1}{x_2 – x_1}<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
  3. Intercepts:<\/strong> Understanding the physical meaning of where a graph hits the axes.<\/li>\n\n\n\n
  4. Simultaneous Equations:<\/strong> Finding the point of intersection (the “break-even” point).<\/li>\n\n\n\n
  5. Linear Modeling:<\/strong> Constructing an equation from a word problem.<\/li>\n<\/ol>\n\n\n\n
    \n\n\n\n

    Q1. A phone plan has a monthly fixed cost of $<\/mi>30<\/mn><\/mrow>\\$30<\/annotation><\/semantics><\/math> and a data charge of $<\/mi>5<\/mn><\/mrow>\\$5<\/annotation><\/semantics><\/math> per gigabyte (g<\/mi>g<\/annotation><\/semantics><\/math>). Which equation models the total monthly cost (C<\/mi>C<\/annotation><\/semantics><\/math>)?<\/p>\n\n\n\n

    A. C<\/em>=30g<\/em>+5<\/p>\n\n\n\n

    B. g<\/em>=5C<\/em>+30<\/p>\n\n\n\n

    C. C<\/em>=5g<\/em>+30<\/p>\n\n\n\n

    D. C<\/em>=35g<\/em><\/p>\n\n\n\n

    Hint: Identify the ‘starting value’ (fixed) and the ‘rate of change’ (per unit).<\/p>\n\n\n\n

    Q2. A line passes through the points (<\/mo>2<\/mn>,<\/mo>10<\/mn>)<\/mo><\/mrow>(2, 10)<\/annotation><\/semantics><\/math> and (<\/mo>5<\/mn>,<\/mo>25<\/mn>)<\/mo><\/mrow>(5, 25)<\/annotation><\/semantics><\/math>. What is the gradient (m<\/mi>m<\/annotation><\/semantics><\/math>) of this line?<\/p>\n\n\n\n

    A. 5<\/p>\n\n\n\n

    B. 0.2<\/p>\n\n\n\n

    C. 15<\/p>\n\n\n\n

    D. 3<\/p>\n\n\n\n

    Hint: Use the formula: \u00a0m<\/mi>=<\/mo>y<\/mi>2<\/mn><\/msub>\u2212<\/mo>y<\/mi>1<\/mn><\/msub><\/mrow>x<\/mi>2<\/mn><\/msub>\u2212<\/mo>x<\/mi>1<\/mn><\/msub><\/mrow><\/mfrac><\/mrow>m = \\frac{y_2 – y_1}{x_2 – x_1}<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

    Q3. What is the x<\/mi>x<\/annotation><\/semantics><\/math>-intercept of the equation 4<\/mn>x<\/mi>+<\/mo>2<\/mn>y<\/mi>=<\/mo>20<\/mn><\/mrow>4x + 2y = 20<\/annotation><\/semantics><\/math>?<\/p>\n\n\n\n

    A. 5<\/p>\n\n\n\n

    B. 10<\/p>\n\n\n\n

    C. 20<\/p>\n\n\n\n

    D. 4<\/p>\n\n\n\n

    Hint: The x<\/em>-intercept occurs when the y<\/em> value is zero.<\/p>\n\n\n\n

    Q4. Solve for x<\/mi>x<\/annotation><\/semantics><\/math> in the equation: 3<\/mn>(<\/mo>x<\/mi>\u2212<\/mo>4<\/mn>)<\/mo>=<\/mo>12<\/mn>+<\/mo>x<\/mi><\/mrow>3(x – 4) = 12 + x<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

    A. 6<\/p>\n\n\n\n

    B. 0<\/p>\n\n\n\n

    C. 24<\/p>\n\n\n\n

    D. 12<\/p>\n\n\n\n

    Hint: Expand the brackets first, then group all x<\/em> terms on one side.<\/p>\n\n\n\n

    Q5. A business has a cost function C<\/mi>=<\/mo>10<\/mn>x<\/mi>+<\/mo>500<\/mn><\/mrow>C = 10x + 500<\/annotation><\/semantics><\/math> and a revenue function R<\/mi>=<\/mo>30<\/mn>x<\/mi><\/mrow>R = 30x<\/annotation><\/semantics><\/math>. How many units (x<\/mi>x<\/annotation><\/semantics><\/math>) must they sell to break even?<\/p>\n\n\n\n

    A. 16.6 units<\/p>\n\n\n\n

    B. 20 units<\/p>\n\n\n\n

    C. 25 units<\/p>\n\n\n\n

    D. 50 units<\/p>\n\n\n\n

    Hint: Break-even occurs when Total Cost equals Total Revenue (C<\/em>=R<\/em>).<\/p>\n\n\n\n

    Q6. If a line has a gradient of \u2212<\/mo>2<\/mn><\/mrow>-2<\/annotation><\/semantics><\/math> and passes through (<\/mo>3<\/mn>,<\/mo>4<\/mn>)<\/mo><\/mrow>(3, 4)<\/annotation><\/semantics><\/math>, what is its y<\/mi>y<\/annotation><\/semantics><\/math>-intercept (c<\/mi>c<\/annotation><\/semantics><\/math>)?<\/p>\n\n\n\n

    A. 2<\/p>\n\n\n\n

    B. \u221210<\/p>\n\n\n\n

    C. 10<\/p>\n\n\n\n

    D. \u22122<\/p>\n\n\n\n

    Hint: Plug the known x<\/em>,y<\/em>, and m<\/em> values into the general linear equation y<\/em>=m<\/em>x<\/em>+c<\/em>.<\/p>\n\n\n\n

    Q7. In a trend analysis of car value, the equation is V<\/mi>=<\/mo>\u2212<\/mo>3000<\/mn>t<\/mi>+<\/mo>45000<\/mn><\/mrow>V = -3000t + 45000<\/annotation><\/semantics><\/math>, where t<\/mi>t<\/annotation><\/semantics><\/math> is years. What does the number \u2212<\/mo>3000<\/mn><\/mrow>-3000<\/annotation><\/semantics><\/math> represent?<\/p>\n\n\n\n

    A. The annual depreciation in dollars.<\/p>\n\n\n\n

    B. The price of the car after one year.<\/p>\n\n\n\n

    C. The total number of cars sold.<\/p>\n\n\n\n

    D. The original price of the car.<\/p>\n\n\n\n

    Hint: Consider what the ‘rate of change’ means in the context of value over time.<\/p>\n\n\n\n

    Q8. Which of the following lines is parallel to y<\/mi>=<\/mo>4<\/mn>x<\/mi>+<\/mo>7<\/mn><\/mrow>y = 4x + 7<\/annotation><\/semantics><\/math>?<\/p>\n\n\n\n

    A. y<\/em>=\u22120.25x<\/em>+7<\/p>\n\n\n\n

    B. y<\/em>=0.25x<\/em>+7<\/p>\n\n\n\n

    C. y<\/em>=\u22124x<\/em>+7<\/p>\n\n\n\n

    D. y<\/em>=4x<\/em>\u221212<\/p>\n\n\n\n

    Hint: Look for the equation with the exact same gradient (m<\/em>) value.<\/p>\n\n\n\n

    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<\/em>=100. This is an example of:<\/p>\n\n\n\n

    A. Interpolation<\/p>\n\n\n\n

    B. Correlation<\/p>\n\n\n\n

    C. Extrapolation<\/p>\n\n\n\n

    D. Simultaneous solving<\/p>\n\n\n\n

    Hint: Is the target value (x<\/mi>=<\/mo>100<\/mn><\/mrow>x=100<\/annotation><\/semantics><\/math>) inside or outside the given data range?<\/p>\n\n\n\n

    Q10. Using the substitution method for y<\/mi>=<\/mo>2<\/mn>x<\/mi><\/mrow>y = 2x<\/annotation><\/semantics><\/math> and x<\/mi>+<\/mo>y<\/mi>=<\/mo>9<\/mn><\/mrow>x + y = 9<\/annotation><\/semantics><\/math> , what is the value of x<\/mi>x<\/annotation><\/semantics><\/math>?<\/p>\n\n\n\n

    A. 9<\/p>\n\n\n\n

    B. 3<\/p>\n\n\n\n

    C. 6<\/p>\n\n\n\n

    D. 4.5<\/p>\n\n\n\n

    Hint: Replace the y<\/em> in the second equation with the expression ‘2x’.<\/p>\n\n\n\n

    ****************************************************************************<\/p>\n\n\n\n

    \u7b2c3\u7ae0\u3067\u306f\u3001\u300c\u50be\u5411\u306e\u8a00\u8a9e\u300d\u3092\u6271\u3044\u307e\u3059\u3002\u4e00\u822c\u6570\u5b66\u306b\u304a\u3044\u3066\u3001\u7dda\u5f62\u65b9\u7a0b\u5f0f\u306f\u5358\u306bx<\/mi>x<\/annotation><\/semantics><\/math>\u3092\u6c42\u3081\u308b\u3060\u3051\u3067\u306a\u304f\u3001\u3042\u308b\u5909\u6570\uff08\u6642\u9593\u306a\u3069\uff09\u304c\u5225\u306e\u5909\u6570\uff08\u8cbb\u7528\u3084\u8ddd\u96e2\u306a\u3069\uff09\u306b\u3069\u306e\u3088\u3046\u306b\u5f71\u97ff\u3059\u308b\u304b\u3092\u30e2\u30c7\u30eb\u5316\u3057\u307e\u3059\u3002\u3053\u306e\u30bb\u30af\u30b7\u30e7\u30f3\u306e\u512a\u79c0\u306a\u751f\u5f92\u306f\u3001\u52fe\u914d\uff08m<\/mi>m<\/annotation><\/semantics><\/math>\uff09\u3092\u300c\u5909\u5316\u7387\u300d\u3068\u3057\u3066\u3001\u307e\u305fy<\/mi>y<\/annotation><\/semantics><\/math>\u5207\u7247\uff08c<\/mi>c<\/annotation><\/semantics><\/math>\uff09\u3092\u300c\u521d\u671f\u6761\u4ef6\u300d\u3068\u3057\u3066\u91cd\u8996\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

    \u6271\u308f\u308c\u308b\u6982\u5ff5\u3068\u30b9\u30ad\u30eb\uff1a<\/p>\n\n\n\n

    \u4ee3\u6570\u7684\u518d\u914d\u7f6e\uff1a\u7dda\u5f62\u30e2\u30c7\u30eb\u306b\u304a\u3051\u308b\u7279\u5b9a\u306e\u5909\u6570\u306e\u89e3\u3092\u6c42\u3081\u308b\u3002<\/p>\n\n\n\n

    \u52fe\u914d\uff08m<\/mi>m<\/annotation><\/semantics><\/math>\uff09\uff1am<\/mi>=<\/mo>y<\/mi>2<\/mn><\/msub>\u2212<\/mo>y<\/mi>1<\/mn><\/msub><\/mrow>x<\/mi>2<\/mn><\/msub>\u2212<\/mo>x<\/mi>1<\/mn><\/msub><\/mrow><\/mfrac><\/mrow>m = \\frac{y_2 – y_1}{x_2 – x_1}<\/annotation><\/semantics><\/math>\u3092\u7528\u3044\u3066\u5909\u5316\u7387\u3092\u8a08\u7b97\u3059\u308b\u3002<\/p>\n\n\n\n

    \u5207\u7247\uff1a\u30b0\u30e9\u30d5\u304c\u8ef8\u3068\u4ea4\u308f\u308b\u4f4d\u7f6e\u306e\u7269\u7406\u7684\u306a\u610f\u5473\u3092\u7406\u89e3\u3059\u308b\u3002<\/p>\n\n\n\n

    \u9023\u7acb\u65b9\u7a0b\u5f0f\uff1a\u4ea4\u70b9\uff08\u300c\u640d\u76ca\u5206\u5c90\u70b9\u300d\uff09\u3092\u6c42\u3081\u308b\u3002<\/p>\n\n\n\n

    \u7dda\u5f62\u30e2\u30c7\u30ea\u30f3\u30b0\uff1a\u6587\u7ae0\u984c\u304b\u3089\u65b9\u7a0b\u5f0f\u3092\u69cb\u7bc9\u3059\u308b\u3002<\/p>\n\n\n\n

    ******************<\/p>\n\n\n\n

    Right answer<\/p>\n\n\n\n

    Q1. C. C<\/em>=5g<\/em>+30 The fixed cost is the y<\/em>-intercept (30) and the rate per GB is the gradient (5).<\/p>\n\n\n\n

    Q2. A. 5 m<\/em>=(25\u221210)\/(5\u22122)=15\/3=5.<\/p>\n\n\n\n

    Q3. A. 5 <\/p>\n\n\n\n

    To find the x<\/mi>x<\/annotation><\/semantics><\/math>-intercept, set y<\/mi>=<\/mo>0<\/mn><\/mrow>y = 0<\/annotation><\/semantics><\/math>.4<\/mn>x<\/mi>+<\/mo>2<\/mn>(<\/mo>0<\/mn>)<\/mo>=<\/mo>20<\/mn><\/mspace>\u27f9<\/mo><\/mspace>4<\/mn>x<\/mi>=<\/mo>20<\/mn><\/mspace>\u27f9<\/mo><\/mspace>x<\/mi>=<\/mo>5<\/mn><\/mrow>4x + 2(0) = 20 \\implies 4x = 20 \\implies x = 5<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

    Q4. D. 12 Expand: 3<\/mn>x<\/mi>\u2212<\/mo>12<\/mn>=<\/mo>12<\/mn>+<\/mo>x<\/mi><\/mrow>3x – 12 = 12 + x<\/annotation><\/semantics><\/math>. Rearrange: 2<\/mn>x<\/mi>=<\/mo>24<\/mn><\/mrow>2x = 24<\/annotation><\/semantics><\/math>. Solve: x<\/mi>=<\/mo>12<\/mn><\/mrow>x = 12<\/annotation><\/semantics><\/math>. <\/p>\n\n\n\n

    Q5. C. 25 units Set C<\/mi>=<\/mo>R<\/mi><\/mrow>C = R<\/annotation><\/semantics><\/math>: 10<\/mn>x<\/mi>+<\/mo>500<\/mn>=<\/mo>30<\/mn>x<\/mi><\/mspace>\u27f9<\/mo><\/mspace>20<\/mn>x<\/mi>=<\/mo>500<\/mn><\/mspace>\u27f9<\/mo><\/mspace>x<\/mi>=<\/mo>25<\/mn><\/mrow>10x + 500 = 30x \\implies 20x = 500 \\implies x = 25<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

    Q6. C. 10<\/p>\n\n\n\n

    Substitute into y<\/mi>=<\/mo>m<\/mi>x<\/mi>+<\/mo>c<\/mi><\/mrow>y = mx + c<\/annotation><\/semantics><\/math>: 4<\/mn>=<\/mo>\u2212<\/mo>2<\/mn>(<\/mo>3<\/mn>)<\/mo>+<\/mo>c<\/mi><\/mspace>\u27f9<\/mo><\/mspace>4<\/mn>=<\/mo>\u2212<\/mo>6<\/mn>+<\/mo>c<\/mi><\/mspace>\u27f9<\/mo><\/mspace>c<\/mi>=<\/mo>10<\/mn><\/mrow>4 = -2(3) + c \\implies 4 = -6 + c \\implies c = 10<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

    Q7. A. The annual depreciation in dollars. The gradient represents the rate of change; since it is negative, the value is decreasing by $<\/mi>3,000<\/mn><\/mrow>\\$3,000<\/annotation><\/semantics><\/math> each year.<\/p>\n\n\n\n

    Q8. y<\/mi>=<\/mo>4<\/mn>x<\/mi>\u2212<\/mo>12<\/mn><\/mrow>y = 4x – 12<\/annotation><\/semantics><\/math> Parallel lines must have the same gradient (m<\/mi>=<\/mo>4<\/mn><\/mrow>m=4<\/annotation><\/semantics><\/math>).<\/p>\n\n\n\n

    Q9. C. Extrapolation Predicting outside the range of known x<\/mi>x<\/annotation><\/semantics><\/math>-values (10<\/mn>10<\/annotation><\/semantics><\/math> to 50<\/mn>50<\/annotation><\/semantics><\/math>) is called extrapolation.<\/p>\n\n\n\n

    Q10. B. 3<\/p>\n\n\n\n

    Substitute 2<\/mn>x<\/mi><\/mrow>2x<\/annotation><\/semantics><\/math> for y<\/mi>y<\/annotation><\/semantics><\/math> in the second equation: x<\/mi>+<\/mo>2<\/mn>x<\/mi>=<\/mo>9<\/mn><\/mspace>\u27f9<\/mo><\/mspace>3<\/mn>x<\/mi>=<\/mo>9<\/mn><\/mspace>\u27f9<\/mo><\/mspace>x<\/mi>=<\/mo>3<\/mn><\/mrow>x + 2x = 9 \\implies 3x = 9 \\implies x = 3<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

    <\/p>\n","protected":false},"excerpt":{"rendered":"

    For Chapter 3, we move into the “language of trends.” In General Mathematics, linear equations aren’t just about finding xx; 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 (mm) as a “rate of change” and the yy-intercept (cc) as […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-972","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/972","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=972"}],"version-history":[{"count":4,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/972\/revisions"}],"predecessor-version":[{"id":977,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/972\/revisions\/977"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=972"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=972"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=972"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}