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1. slope = 5, through (3,13) 2. through (1,1), (0,4) 3. slope = $-\\fra…

Question

  1. slope = 5, through (3,13)
  2. through (1,1), (0,4)
  3. slope = $-\frac{5}{3}$, through (3, -1)
  4. through (1, 3), (-5, -15)
  5. slope = -4, through (-2, 9)
  6. through (2, -1), (3, -3)
  7. slope = -2, through (-4, -2)
  8. through (1, -4), (-2, 5)
  9. slope = $\frac{1}{3}$, through (6, 9)
  10. through (-4, 1), (5, -2)
  11. slope = $-\frac{1}{4}$, through (-2, 6)
  12. through (-3, -2), (5, -2)
  13. undefined slope through (5, 2)
  14. through (3, 6), (3, -1)
  15. \
$$\begin{tabular}{c|c|c|c|c|c} $x$ & -2 & -1 & 0 & 1 & 2 \\\\ $y$ & -3 & -1 & 1 & 3 & 5 \\\\ \\end{tabular}$$
  1. graph of a line with y-axis labeled 50, 100, 150 and x-axis labeled 50, 100, 150
  2. \
$$\begin{tabular}{c|c|c|c|c|c} $x$ & -4 & -2 & 0 & 2 & 4 \\\\ $y$ & 1 & 2 & 3 & 4 & 5 \\\\ \\end{tabular}$$
  1. \
$$\begin{tabular}{c|c|c|c|c|c} $x$ & 5 & 3 & 0 & 1 & 2 \\\\ $y$ & 17 & 11 & 2 & 5 & 8 \\\\ \\end{tabular}$$
  1. graph of a line with y-axis labeled -900, -600, -300, 0, 300, 600, 900 and x-axis labeled -900, -600, -300, 0, 300, 600, 900

Explanation:

To solve these problems, we will use the point - slope form of a linear equation, which is $y - y_1=m(x - x_1)$, where $m$ is the slope and $(x_1,y_1)$ is a point on the line. For problems where we are given two points, we first find the slope using the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$ and then use the point - slope form.

Problem 1: slope $ = 5$, through $(3,13)$

Step 1: Identify the values

We know that $m = 5$, $x_1=3$ and $y_1 = 13$.

Step 2: Substitute into point - slope form

Using the formula $y - y_1=m(x - x_1)$, we substitute the values:
$y-13 = 5(x - 3)$

Step 3: Simplify the equation

Expand the right - hand side: $y-13=5x-15$
Add 13 to both sides: $y=5x - 15 + 13$
$y=5x-2$

Problem 2: through $(1,1),(0,4)$

Step 1: Find the slope

Using the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$, let $(x_1,y_1)=(1,1)$ and $(x_2,y_2)=(0,4)$. Then $m=\frac{4 - 1}{0 - 1}=\frac{3}{-1}=-3$

Step 2: Use point - slope form

We can use the point $(0,4)$ (since $x_1 = 0,y_1 = 4$ and $m=-3$). The point - slope form is $y - y_1=m(x - x_1)$. Substituting the values, we get $y - 4=-3(x - 0)$

Step 3: Simplify the equation

$y-4=-3x$
Add 4 to both sides: $y=-3x + 4$

Problem 3: slope $=-\frac{5}{3}$, through $(3,-1)$

Step 1: Identify the values

We have $m =-\frac{5}{3}$, $x_1 = 3$ and $y_1=-1$

Step 2: Substitute into point - slope form

Using $y - y_1=m(x - x_1)$, we substitute: $y-(-1)=-\frac{5}{3}(x - 3)$

Step 3: Simplify the equation

$y + 1=-\frac{5}{3}x+5$
Subtract 1 from both sides: $y=-\frac{5}{3}x+4$

Problem 4: through $(1,3),(-5,-15)$

Step 1: Find the slope

Using $m=\frac{y_2 - y_1}{x_2 - x_1}$, with $(x_1,y_1)=(1,3)$ and $(x_2,y_2)=(-5,-15)$, we have $m=\frac{-15 - 3}{-5 - 1}=\frac{-18}{-6}=3$

Step 2: Use point - slope form

Using the point $(1,3)$ ( $x_1 = 1,y_1 = 3$ and $m = 3$), the point - slope form gives $y - 3=3(x - 1)$

Step 3: Simplify the equation

$y-3=3x-3$
Add 3 to both sides: $y=3x$

Problem 5: slope $=-4$, through $(-2,9)$

Step 1: Identify the values

$m=-4$, $x_1=-2$ and $y_1 = 9$

Step 2: Substitute into point - slope form

Using $y - y_1=m(x - x_1)$, we get $y - 9=-4(x+2)$

Step 3: Simplify the equation

Expand the right - hand side: $y - 9=-4x-8$
Add 9 to both sides: $y=-4x + 1$

Problem 6: through $(2,-1),(3,-3)$

Step 1: Find the slope

Using $m=\frac{y_2 - y_1}{x_2 - x_1}$, with $(x_1,y_1)=(2,-1)$ and $(x_2,y_2)=(3,-3)$, we have $m=\frac{-3+1}{3 - 2}=\frac{-2}{1}=-2$

Step 2: Use point - slope form

Using the point $(2,-1)$ ( $x_1 = 2,y_1=-1$ and $m=-2$), the point - slope form is $y+1=-2(x - 2)$

Step 3: Simplify the equation

$y + 1=-2x + 4$
Subtract 1 from both sides: $y=-2x + 3$

Problem 7: slope $=-2$, through $(-4,-2)$

Step 1: Identify the values

$m=-2$, $x_1=-4$ and $y_1=-2$

Step 2: Substitute into point - slope form

Using $y - y_1=m(x - x_1)$, we get $y+2=-2(x + 4)$

Step 3: Simplify the equation

Expand the right - hand side: $y + 2=-2x-8$
Subtract 2 from both sides: $y=-2x-10$

Problem 8: through $(1,-4),(-2,5)$

Step 1: Find the slope

Using $m=\frac{y_2 - y_1}{x_2 - x_1}$, with $(x_1,y_1)=(1,-4)$ and $(x_2,y_2)=(-2,5)$, we have $m=\frac{5 + 4}{-2 - 1}=\frac{9}{-3}=-3$

Step 2: Use point - slope form

Using the point $(1,-4)$ ( $x_1 = 1,y_1=-4$ and $m=-3$), the point - slope form is $y + 4=-3(x - 1)$

Step 3: Simplify the equation

$y + 4=-3x+3$
Subtract 4 from both sides: $y=-3x-1$

Problem 9: slope $=\frac{1}{3}$, through $(6,9)$

Step 1: Identify the values

$m=\frac{1}{3}$, $x_1 = 6$ and $y_1 = 9$

Step 2: Substitute…

Answer:

s:

  1. $y = 5x-2$
  2. $y=-3x + 4$
  3. $y=-\frac{5}{3}x + 4$
  4. $y = 3x$
  5. $y=-4x + 1$
  6. $y=-2x + 3$
  7. $y=-2x-10$
  8. $y=-3x-1$
  9. $y=\frac{1}{3}x + 7$
  10. $y=-\frac{1}{3}x-\frac{1}{3}$
  11. $y=-\frac{1}{4}x+\frac{11}{2}$
  12. $y=-2$
  13. $x = 5$
  14. $x = 3$
  15. $y=2x + 1$
  16. $y=\frac{1}{2}x+50$ (assuming the points on the graph)
  17. $y=\frac{1}{2}x+3$
  18. $y=3x + 2$
  19. $y=-x$ (assuming the points on the graph)