This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Part 1: Find the gradient, y-intercept, and x-intercept for each equation. a) y = -(2x - 2) Step 1: Simplify the equation to the slope-intercept form y = mx + c. y = -2x + 2 Step 2: Identify the gradient, y-intercept, and x-intercept. The gradient m is the coefficient of x. The y-intercept c is the constant term. To find the x-intercept, set y = 0 and solve for x. 0 = -2x + 2 2x = 2 x = 1 • Gradient: -2 • Y-intercept: 2 • X-intercept: 1 b) y - (1)/(4)x - 6 = 0 Step 1: Rearrange the equation to the slope-intercept form y = mx + c. y = (1)/(4)x + 6 Step 2: Identify the gradient, y-intercept, and x-intercept. To find the x-intercept, set y = 0 and solve for x. 0 = (1)/(4)x + 6 -(1)/(4)x = 6 x = 6 × (-4) x = -24 • Gradient: (1)/(4) • Y-intercept: 6 • X-intercept: -24 c) -3x + y = 2 Step 1: Rearrange the equation to the slope-intercept form y = mx + c. y = 3x + 2 Step 2: Identify the gradient, y-intercept, and x-intercept. To find the x-intercept, set y = 0 and solve for x. 0 = 3x + 2 -3x = 2 x = -(2)/(3) • Gradient: 3 • Y-intercept: 2 • X-intercept: -(2)/(3) d) 2 + y = x Step 1: Rearrange the equation to the slope-intercept form y = mx + c. y = x - 2 Step 2: Identify the gradient, y-intercept, and x-intercept. To find the x-intercept, set y = 0 and solve for x. 0 = x - 2 x = 2 • Gradient: 1 • Y-intercept: -2 • X-intercept: 2 e) -6x + y = 2 Step 1: Rearrange the equation to the slope-intercept form y = mx + c. y = 6x + 2 Step 2: Identify the gradient, y-intercept, and x-intercept. To find the x-intercept, set y = 0 and solve for x. 0 = 6x + 2 -6x = 2 x = -(2)/(6) x = -(1)/(3) • Gradient: 6 • Y-intercept: 2 • X-intercept: -(1)/(3) f) (6 - 2y)/(3) = 2 Step 1: Simplify and rearrange the equation. Multiply both sides by 3: 6 - 2y = 6 Subtract 6 from both sides: -2y = 0 Divide by -2: y = 0 Step 2: Identify the gradient, y-intercept, and x-intercept. This is a horizontal line along the x-axis. • Gradient: 0 • Y-intercept: 0 • X-intercept: All real numbers g) -(x - 2y) - (-x - 2y) = -2 Step 1: Simplify the equation. -x + 2y + x + 2y = -2 4y = -2 y = -(2)/(4) y = -(1)/(2) Step 2: Identify the gradient, y-intercept, and x-intercept. This is a horizontal line. • Gradient: 0 • Y-intercept: -(1)/(2) • X-intercept: None h) (x + 1)/(4 + 1) = (3)/(2 + 2) Step 1: Simplify the equation. (x + 1)/(5) = (3)/(4) Multiply both sides by 5: x + 1 = (15)/(4) Subtract 1 from both sides: x = (15)/(4) - 1 x = (15)/(4) - (4)/(4) x = (11)/(4) Step 2: Identify the gradient, y-intercept, and x-intercept. This is a vertical line. • Gradient: Undefined • Y-intercept: None • X-intercept: (11)/(4) Part 2: Find the equations of lines parallel to x - 2y = 6 that pass through each of the following points. Step 1: Find the gradient of the given line x - 2y = 6. Rearrange the equation into slope-intercept form y = mx + c: -2y = -x + 6 y = (1)/(2)x - 3 The gradient of this line is m = (1)/(2). Since the new lines are parallel, their gradient will also be m = (1)/(2). a) Point: (-4, 1) Step 2: Use the point-slope form y - y_1 = m(x - x_1) with m = (1)/(2) and (x_1, y_1) = (-4, 1). y - 1 = (1)/(2)(x - (-4)) y - 1 = (1)/(2)(x + 4) Step 3: Simplify the equation to slope-intercept form. y - 1 = (1)/(2)x + 2 y = (1)/(2)x + 3 The equation of the line is y = (1)/(2)x + 3. b) Point: ((1)/(2), 0) Step 2: Use the point-slope form y - y_1 = m(x - x_1) with m = (1)/(2) and (x_1, y_1) = ((1)/(2), 0). y - 0 = (1)/(2)(x - (1)/(2)) Step 3: Simplify the equation to slope-intercept form. y = (1)/(2)x - (1)/(4) The equation of the line is y = (1)/(2)x - (1)/(4). Got more? Send 'em 📸