Solved Exercises on Two-Variable Linear Equations in Integrated Math 1

Master two-variable linear equations: standard form, slope-intercept form, graphing, and solution methods through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Standard Form Problem
Exercise 1
Rewrite the equation \(y = 2x + 5\) in standard form \(Ax + By = C\). Then find two ordered pairs that satisfy the equation.
Definition:

Standard Form: A linear equation in the form \(Ax + By = C\) where \(A\), \(B\), and \(C\) are integers, and \(A\) is non-negative

Standard Form Conversion Method:
  1. Move all variables to one side of the equation
  2. Move the constant to the other side
  3. Ensure the coefficient of \(x\) is positive
  4. Verify that coefficients are integers
Starting Equation
\(y = 2x + 5\)
Rearrange Terms
\(-2x + y = 5\)
Multiply by -1
\(2x - y = -5\)
Step 1: Start with slope-intercept form

\(y = 2x + 5\)

Step 2: Move all terms to one side

Subtract \(2x\) from both sides: \(-2x + y = 5\)

Step 3: Ensure positive coefficient for x

Multiply by \(-1\): \(2x - y = -5\)

Step 4: Find two ordered pairs

Let \(x = 0\): \(2(0) - y = -5\) → \(-y = -5\) → \(y = 5\), so \((0, 5)\)

Let \(x = 1\): \(2(1) - y = -5\) → \(2 - y = -5\) → \(y = 7\), so \((1, 7)\)

Standard form: \(2x - y = -5\)
Final answer:

Standard form: \(2x - y = -5\), Solutions: \((0, 5)\) and \((1, 7)\)

Applied rules:

• Standard Form: \(Ax + By = C\) with integer coefficients and \(A \geq 0\)

• Equation Manipulation: Perform identical operations on both sides

• Ordered Pairs: Substitute values to verify solutions

2 Graphing Problem
Exercise 2
Graph the equation \(3x + 2y = 6\) by finding the x-intercept and y-intercept, then draw the line.
Definition:

Intercepts: The x-intercept occurs when \(y = 0\), and the y-intercept occurs when \(x = 0\)

X-intercept
Set y=0: (2,0)
Y-intercept
Set x=0: (0,3)
Draw Line
Connect intercepts
Step 1: Find x-intercept (where y = 0)

\(3x + 2(0) = 6\) → \(3x = 6\) → \(x = 2\), so \((2, 0)\)

Step 2: Find y-intercept (where x = 0)

\(3(0) + 2y = 6\) → \(2y = 6\) → \(y = 3\), so \((0, 3)\)

Step 3: Plot the intercepts and draw the line

Plot points \((2, 0)\) and \((0, 3)\), then connect with a straight line

Step 4: Verify with a third point

Let \(x = 1\): \(3(1) + 2y = 6\) → \(3 + 2y = 6\) → \(y = 1.5\), so \((1, 1.5)\)

Intercepts: (2,0) and (0,3)
Final answer:

The line passes through \((2, 0)\) and \((0, 3)\) with equation \(3x + 2y = 6\)

Applied rules:

• Intercept Method: Use x-intercept and y-intercept to graph linear equations

• Linearity: Two points determine a unique line

• Verification: Check with additional points

3 Solution Verification Problem
Exercise 3
Determine if the ordered pair \((-1, 4)\) is a solution to the equation \(2x - 3y = -14\). Also find the value of y when x = 5.
Definition:

Solution to Linear Equation: An ordered pair \((x, y)\) that makes the equation true when substituted

Check Point
2(-1)-3(4)=-14
Substitute x=5
2(5)-3y=-14
Solve for y
y=8
Step 1: Check if \((-1, 4)\) satisfies the equation

Substitute \(x = -1\) and \(y = 4\) into \(2x - 3y = -14\)

\(2(-1) - 3(4) = -2 - 12 = -14\) ✓

Step 2: Find y when x = 5

Substitute \(x = 5\) into \(2x - 3y = -14\)

\(2(5) - 3y = -14\) → \(10 - 3y = -14\) → \(-3y = -24\) → \(y = 8\)

Step 3: Verify the new solution

Check \((5, 8)\): \(2(5) - 3(8) = 10 - 24 = -14\) ✓

(-1,4) is a solution, when x=5, y=8
Final answer:

Yes, \((-1, 4)\) is a solution. When \(x = 5\), \(y = 8\)

Applied rules:

• Solution Verification: Substitute coordinates into equation

• Algebraic Substitution: Replace variables with known values

• Isolation: Solve for unknown variable by isolating it

Linear Equation Forms and Properties
\(Ax + By = C\)
Standard Form
Standard Form
\(Ax + By = C\)
General form with A ≥ 0
Slope-Intercept
\(y = mx + b\)
m=slope, b=y-intercept
Point-Slope
\(y - y_1 = m(x - x_1)\)
Through point (x₁,y₁)
Key definitions:

Linear Equation: An equation that forms a straight line when graphed

Ordered Pair: A pair of numbers \((x, y)\) that represents a point on the coordinate plane

Solution: An ordered pair that makes the equation true when substituted

Linear Equation Methodology:
  1. Identify: Determine the form of the equation
  2. Manipulate: Convert between forms as needed
  3. Solve: Find specific solutions or graph the equation
  4. Verify: Check solutions by substitution
Tip 1: In standard form, A, B, and C should be integers, and A should be non-negative.
Tip 2: To convert to slope-intercept form, solve for y.
Tip 3: Always verify solutions by substituting back into the original equation.
Tip 4: A linear equation has infinitely many solutions, forming a line when graphed.
Common Mistakes: Forgetting to change signs when rearranging equations, arithmetic errors, misidentifying coefficients.
Memorization Tip: Remember that every point on the line is a solution to the equation.
Solution: Exercises 4 to 5
4 Form Conversion Problem
Exercise 4
Convert the equation \(y = -\frac{3}{4}x + 2\) to standard form. Then identify the slope and y-intercept.
Definition:

Slope-Intercept Form: \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept

Starting Equation
\(y = -\frac{3}{4}x + 2\)
Eliminate Fraction
Multiply by 4
Standard Form
\(3x + 4y = 8\)
Step 1: Start with slope-intercept form

\(y = -\frac{3}{4}x + 2\)

Step 2: Eliminate the fraction by multiplying by denominator

Multiply everything by 4: \(4y = -3x + 8\)

Step 3: Rearrange to standard form

Add \(3x\) to both sides: \(3x + 4y = 8\)

Step 4: Identify slope and y-intercept from original form

From \(y = -\frac{3}{4}x + 2\): slope \(m = -\frac{3}{4}\), y-intercept \(b = 2\)

Standard form: \(3x + 4y = 8\)
Final answer:

Standard form: \(3x + 4y = 8\), Slope: \(m = -\frac{3}{4}\), Y-intercept: \(b = 2\)

Applied rules:

• Fraction Elimination: Multiply by LCD to clear fractions

• Standard Form: Ensure integer coefficients with A ≥ 0

• Slope-Intercept Recognition: Coefficient of x is slope, constant term is y-intercept

5 Table of Values Problem
Exercise 5
Complete the table of values for the equation \(x - 2y = 4\) and use it to sketch the graph.
Definition:

Table of Values: A collection of ordered pairs that satisfy a linear equation

Given Equation
\(x - 2y = 4\)
Solve for y
\(y = \frac{x-4}{2}\)
Complete Table
Select x values
Step 1: Solve for y to make calculations easier

\(x - 2y = 4\) → \(-2y = -x + 4\) → \(y = \frac{x-4}{2}\)

Step 2: Choose convenient x values and calculate corresponding y values

When \(x = 0\): \(y = \frac{0-4}{2} = -2\), so \((0, -2)\)

When \(x = 2\): \(y = \frac{2-4}{2} = -1\), so \((2, -1)\)

When \(x = 4\): \(y = \frac{4-4}{2} = 0\), so \((4, 0)\)

When \(x = 6\): \(y = \frac{6-4}{2} = 1\), so \((6, 1)\)

Step 3: Verify one point

Check \((4, 0)\): \(4 - 2(0) = 4 - 0 = 4\) ✓

Step 4: Plot the points and draw the line

Plot \((0, -2)\), \((2, -1)\), \((4, 0)\), and \((6, 1)\), then connect with a straight line

Points: (0,-2), (2,-1), (4,0), (6,1)
Final answer:

Table includes points \((0, -2)\), \((2, -1)\), \((4, 0)\), and \((6, 1)\). The line passes through these points.

Applied rules:

• Table Completion: Select convenient x-values and solve for y

• Graphing: Plot points from table and connect with straight line

• Verification: Check at least one point in the original equation

Linear Equations Summary: Definitions, Rules, Methods
\(y = mx + b \leftrightarrow Ax + By = C\)
Form Conversion
Key definitions:

Linear Equation in Two Variables: An equation that can be written in the form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants and not both zero

Solution: An ordered pair \((x, y)\) that makes the equation true when substituted

Slope: The rate of change, calculated as rise over run (\(m = \frac{y_2-y_1}{x_2-x_1}\))

Intercepts: Points where the graph crosses the axes (x-intercept: y=0, y-intercept: x=0)

Complete Linear Equation Methodology:
  1. Identification: Recognize the form of the linear equation
  2. Conversion: Change between standard form, slope-intercept form, and point-slope form
  3. Solution Finding: Determine specific ordered pairs that satisfy the equation
  4. Graphing: Plot the line using intercepts, tables, or slope-intercept form
  5. Verification: Confirm solutions by substitution
Tip 1: To convert from slope-intercept to standard form, move x-term to the left side.
Tip 2: Always clear fractions by multiplying by the LCD when converting to standard form.
Tip 3: When finding intercepts, set the other variable to zero.
Tip 4: Linear equations have infinitely many solutions forming a straight line.
Common Errors: Sign errors when rearranging equations, arithmetic mistakes with fractions, forgetting to verify solutions.
Exam Preparation: Practice form conversion, master intercept method, understand slope concepts thoroughly.
Essential Forms and Properties:

• Standard Form: \(Ax + By = C\) (A, B, C integers, A ≥ 0)

• Slope-Intercept: \(y = mx + b\) (m = slope, b = y-intercept)

• Point-Slope: \(y - y_1 = m(x - x_1)\) (through point \((x_1, y_1)\))

• Horizontal Line: \(y = c\) (slope = 0)

• Vertical Line: \(x = c\) (slope undefined)

Questions & Answers

Question: I'm confused about when to use standard form vs. slope-intercept form. How do I know which one to use?

Answer: The choice depends on what you're trying to accomplish:

  • Standard Form (Ax + By = C): Best for finding intercepts, working with integer coefficients, or when the equation is naturally in this form
  • Slope-Intercept Form (y = mx + b): Best for graphing quickly, identifying slope and y-intercept, or solving for y
  • Point-Slope Form: Best when you know a point and the slope

As a general rule, convert to slope-intercept form when you need to graph or identify slope/y-intercept. Use standard form when you're looking for intercepts or need integer coefficients.

Question: Why do we need to make sure A is positive in standard form? Does it really matter?

Answer: It's a convention that ensures standard form is truly "standard." When A is positive:

  • All equations have a consistent format
  • It makes comparison between equations easier
  • It simplifies identification of intercepts and other properties

While the equation is still mathematically correct with a negative A, the standard convention is to make A positive. This is achieved by multiplying the entire equation by -1 if necessary.

Question: How can I check if I've correctly converted an equation from one form to another?

Answer: Here are effective verification methods:

  • Algebraic Check: Work backwards from your converted form to see if you get the original
  • Numerical Check: Pick a value for x, substitute into both forms, and verify you get the same y-value
  • Graphical Check: Both forms should produce the same line when graphed
  • Substitution Check: Use the same point in both equations to ensure they both hold true

For example, if you convert y = 2x + 3 to 2x - y = -3, pick x = 1: in first form y = 5, in second form 2(1) - 5 = -3 ✓