Q1 Solve by substitution

Question

y=4x-9
y=x-3
Q2 Solve by substitution
4x-y=20
y=-x+5

answer both with simple to the point equations showing your answer

Answer 1

Sure! Let's solve both systems of equations by substitution.

Q1:

Equations:

1. \( y = 4x - 9 \)
2. \( y = x - 3 \)

Step 1: Set the equations equal to each other.
Since both equations equal \( y \), we can set them equal to each other:
\[ 4x - 9 = x - 3 \]

Step 2: Solve for \( x \).
\[ 4x - x = -3 + 9 \]
\[ 3x = 6 \]
\[ x = 2 \]

Step 3: Substitute \( x \) back into one of the original equations to find \( y \).
Using \( y = x - 3 \):
\[ y = 2 - 3 \]
\[ y = -1 \]

Solution for Q1:
\( (x, y) = (2, -1) \)

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Q2:

Equations:

1. \( 4x - y = 20 \)
2. \( y = -x + 5 \)

Step 1: Substitute the expression for \( y \) from the second equation into the first equation.
\[ 4x - (-x + 5) = 20 \]
\[ 4x + x - 5 = 20 \]
\[ 5x - 5 = 20 \]

Step 2: Solve for \( x \).
\[ 5x = 20 + 5 \]
\[ 5x = 25 \]
\[ x = 5 \]

Step 3: Substitute \( x \) back into the second equation to find \( y \).
Using \( y = -x + 5 \):
\[ y = -5 + 5 \]
\[ y = 0 \]

Solution for Q2:
\( (x, y) = (5, 0) \)

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Final Answers:

• Q1: \( (2, -1) \)
• Q2: \( (5, 0) \)

Answer 2

Q3 Solve the system by graphing. the graph is a grid with x and y axis, both reaching out till 5/-5
y=-3x+4
y=3x-2

Answer 3

To solve the system of equations by graphing, we will graph the lines represented by the equations:

1. \( y = -3x + 4 \)
2. \( y = 3x - 2 \)

Step 1: Find Points for Each Equation

For \( y = -3x + 4 \):

• When \( x = 0 \):
  \( y = -3(0) + 4 = 4 \)
  Point: \( (0, 4) \)

• When \( x = 1 \):
  \( y = -3(1) + 4 = 1 \)
  Point: \( (1, 1) \)

• When \( x = 2 \):
  \( y = -3(2) + 4 = -2 \)
  Point: \( (2, -2) \)

For \( y = 3x - 2 \):

• When \( x = 0 \):
  \( y = 3(0) - 2 = -2 \)
  Point: \( (0, -2) \)

• When \( x = 1 \):
  \( y = 3(1) - 2 = 1 \)
  Point: \( (1, 1) \)

• When \( x = 2 \):
  \( y = 3(2) - 2 = 4 \)
  Point: \( (2, 4) \)

Step 2: Plot the Points on the Graph

1. For \( y = -3x + 4 \), plot the points:

   • \( (0, 4) \)
   • \( (1, 1) \)
   • \( (2, -2) \)

2. For \( y = 3x - 2 \), plot the points:

   • \( (0, -2) \)
   • \( (1, 1) \) (this point is common to both lines)
   • \( (2, 4) \)

Step 3: Draw the Lines

• Connect the points for \( y = -3x + 4 \) with a straight line.
• Connect the points for \( y = 3x - 2 \) with a straight line.

Step 4: Identify the Intersection

From the plotting, you'll see that both lines intersect at the point \( (1, 1) \).

Final Answer

The solution to the system of equations is:
\( (x, y) = (1, 1) \)

Graph visualization

If you were to visualize the graph, it would show two lines crossing at the point (1, 1). If graphed correctly on a grid with \( x \) and \( y \) axes reaching from \( -5 \) to \( 5 \), you would see each line's slope and points correctly plotted based on the calculations.

Answer 4

Q4 Sketch the solution to the system of inequalities. same grid as previous question
y-52x-2
y<-12x+2

Answer 5

To sketch the solution to the system of inequalities, we will first rewrite the two inequalities given and then graph them:

1. First Inequality:
   \( y \geq 5x - 2 \)
   (Note: I corrected the notation from \( y - 52x - 2 \) to \( y \geq 5x - 2 \) based on a typical format for inequalities. If this was not the case, please clarify.)

2. Second Inequality:
   \( y < -12x + 2 \)

Step 1: Graph the Boundary Lines

For \( y = 5x - 2 \) (which is the boundary for the first inequality):

• Find the y-intercept (when \( x = 0 \)):
  \( y = 5(0) - 2 = -2 \)
  Point: \( (0, -2) \)

• Find another point (when \( x = 1 \)):
  \( y = 5(1) - 2 = 3 \)
  Point: \( (1, 3) \)

• Graph the line through the points \( (0, -2) \) and \( (1, 3) \). Since the inequality is \( \geq \), this line will be solid.

For \( y = -12x + 2 \) (which is the boundary for the second inequality):

• Find the y-intercept (when \( x = 0 \)):
  \( y = -12(0) + 2 = 2 \)
  Point: \( (0, 2) \)

• Find another point (when \( x = 1 \)):
  \( y = -12(1) + 2 = -10 \)
  Point: \( (1, -10) \)

• Graph the line through the points \( (0, 2) \) and \( (1, -10) \). Since the inequality is \( < \), this line will be dashed.

Step 2: Determine the Shading for Each Inequality

• For \( y \geq 5x - 2 \):
  Shade the area above the solid line because we are looking for values of \( y \) that are greater than or equal to the line.

• For \( y < -12x + 2 \):
  Shade the area below the dashed line because we are looking for values of \( y \) that are less than the line.

Step 3: Identify the Solution Region

The solution to the system of inequalities will be where the shaded areas overlap.

Final Representation

• The solid line \( y = 5x - 2 \) indicates the values that are included, while the dashed line \( y = -12x + 2 \) indicates the values that are excluded.
• The final graph will show the intersection of the shaded areas.

Quick Sketch Example (textual):

• Draw the x and y axes with a range from -5 to 5.
• Plot the solid line from the points \( (0, -2) \) to \( (1, 3) \).
• Plot the dashed line from the points \( (0, 2) \) to \( (1, -10) \).
• Shade above the solid line and below the dashed line.

You would see an area that typically looks like an angular region between the two lines, showcasing where the two inequalities coexist. If you have graph paper or a graphing tool, you can visualize this intersection accurately!

Answer 6

not a sketch but an answer

Answer 7

I apologize for misunderstanding your request earlier. Let's provide a clear answer to the inequalities without the sketching details.

Given Inequalities:

1. \( y \geq 5x - 2 \)
2. \( y < -12x + 2 \)

Step 1: Identify the Boundary Lines

1. Boundary Line for \( y \geq 5x - 2 \):

   • This line includes points where \( y \) is equal to \( 5x - 2 \). Since this is a "greater than or equal to" inequality, the line will be solid.

2. Boundary Line for \( y < -12x + 2 \):

   • This line includes points where \( y \) is equal to \( -12x + 2 \). Since this is a "less than" inequality, the line will be dashed.

Step 2: Determine the Shading Regions

• For the inequality \( y \geq 5x - 2 \): Shade above the line (including the line).
• For the inequality \( y < -12x + 2 \): Shade below the line (not including the line).

Step 3: Identify the Solution Region

The solution to the system of inequalities is the region where the shaded areas overlap:

• The solution set consists of all points \( (x, y) \) that are:
  • Above the solid line \( y = 5x - 2 \)
  • Below the dashed line \( y = -12x + 2 \)

Final Answer:

The solution to the system of inequalities is the set of all points \( (x, y) \) that satisfy:

• \( y \geq 5x - 2 \)
• \( y < -12x + 2 \)

This region represents the intersection of the two shaded areas on the graph.

Please note that you would typically express the solution region using set notation if needed, indicating the conditions that define the overlapping area.