Solved Exercises on X and Y Intercepts in Integrated Math 1

Master x and y intercepts: finding intercepts, graphing lines, and applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Intercept Finding
Exercise 1
Find the x-intercept and y-intercept of the equation \(2x + 3y = 12\). Then graph the line using these intercepts.
Definition:

X-intercept: The point where a graph crosses the x-axis (y = 0). The coordinate is \((a, 0)\).

Y-intercept: The point where a graph crosses the y-axis (x = 0). The coordinate is \((0, b)\).

Intercept Finding Method:
  1. To find x-intercept: Set y = 0 and solve for x
  2. To find y-intercept: Set x = 0 and solve for y
  3. Plot both intercepts and draw the line connecting them
X-intercept
Set y=0: 2x=12
Y-intercept
Set x=0: 3y=12
Results
(6,0) and (0,4)
Step 1: Find x-intercept (set y = 0)

\(2x + 3(0) = 12\) → \(2x = 12\) → \(x = 6\)

X-intercept: \((6, 0)\)

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

\(2(0) + 3y = 12\) → \(3y = 12\) → \(y = 4\)

Y-intercept: \((0, 4)\)

Step 3: Plot the intercepts and draw the line

Plot \((6, 0)\) and \((0, 4)\) on the coordinate plane, then connect with a straight line

Step 4: Verify with a third point

Let \(x = 3\): \(2(3) + 3y = 12\) → \(6 + 3y = 12\) → \(y = 2\), so \((3, 2)\)

X-int: (6,0), Y-int: (0,4)
Final answer:

X-intercept: \((6, 0)\), Y-intercept: \((0, 4)\). The line passes through these points.

Applied rules:

• Intercept Definition: Set opposite variable to zero

• Algebraic Solving: Isolate the remaining variable

• Graphing: Two points determine a unique line

2 Slope-Intercept Form
Exercise 2
Find the x-intercept and y-intercept of the equation \(y = -\frac{1}{2}x + 3\). Identify which form this equation is in and explain how to find the y-intercept directly.
Definition:

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

Y-intercept
Directly: (0,3)
X-intercept
Set y=0: 0=-½x+3
Result
(6,0) and (0,3)
Step 1: Identify the form

This equation is in slope-intercept form: \(y = mx + b\) where \(m = -\frac{1}{2}\) and \(b = 3\)

Step 2: Find y-intercept directly

In slope-intercept form, the y-intercept is the constant term: \((0, 3)\)

Step 3: Find x-intercept (set y = 0)

\(0 = -\frac{1}{2}x + 3\) → \(\frac{1}{2}x = 3\) → \(x = 6\)

X-intercept: \((6, 0)\)

Step 4: Verify the intercepts

Check \((6, 0)\): \(0 = -\frac{1}{2}(6) + 3 = -3 + 3 = 0\) ✓

Check \((0, 3)\): \(3 = -\frac{1}{2}(0) + 3 = 0 + 3 = 3\) ✓

X-int: (6,0), Y-int: (0,3)
Final answer:

X-intercept: \((6, 0)\), Y-intercept: \((0, 3)\). The equation is in slope-intercept form.

Applied rules:

• Slope-Intercept Recognition: \(y = mx + b\) form reveals y-intercept as \((0, b)\)

• X-intercept Finding: Set \(y = 0\) and solve for \(x\)

• Verification: Substitute intercepts back into original equation

3 Fractional Coefficients
Exercise 3
Find the x-intercept and y-intercept of the equation \(\frac{1}{3}x + \frac{2}{5}y = 2\). Show all steps for clearing fractions.
Definition:

Least Common Denominator (LCD): The smallest number that all denominators divide into evenly, used to clear fractions

Original Eq
\(\frac{1}{3}x + \frac{2}{5}y = 2\)
Multiply by LCD
×15: 5x+6y=30
Intercepts
(6,0) and (0,5)
Step 1: Clear fractions by multiplying by LCD

The denominators are 3 and 5, so LCD = 15

\(15 \times \left(\frac{1}{3}x + \frac{2}{5}y\right) = 15 \times 2\)

\(5x + 6y = 30\)

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

\(5x + 6(0) = 30\) → \(5x = 30\) → \(x = 6\)

X-intercept: \((6, 0)\)

Step 3: Find y-intercept (set x = 0)

\(5(0) + 6y = 30\) → \(6y = 30\) → \(y = 5\)

Y-intercept: \((0, 5)\)

Step 4: Verify with original equation

Check \((6, 0)\): \(\frac{1}{3}(6) + \frac{2}{5}(0) = 2 + 0 = 2\) ✓

Check \((0, 5)\): \(\frac{1}{3}(0) + \frac{2}{5}(5) = 0 + 2 = 2\) ✓

X-int: (6,0), Y-int: (0,5)
Final answer:

X-intercept: \((6, 0)\), Y-intercept: \((0, 5)\). Multiplying by LCD cleared the fractions.

Applied rules:

• Fraction Clearing: Multiply by LCD to eliminate fractions

• Intercept Finding: Set opposite variable to zero

• Verification: Always check with original equation

Intercept Finding Rules and Methods
\(\text{X-intercept: } (a, 0), \text{ Y-intercept: } (0, b)\)
Intercept Coordinates
X-intercept
Set y = 0
Solve for x
Y-intercept
Set x = 0
Solve for y
Slope-Int Form
y = mx + b
Y-int at (0,b)
Key definitions:

X-intercept: The point where a graph crosses the x-axis (y-coordinate is 0)

Y-intercept: The point where a graph crosses the y-axis (x-coordinate is 0)

Linear Function: A function whose graph is a straight line

Intercept Finding Methodology:
  1. Identify: Determine the form of the equation
  2. Set Variable: Set the opposite variable to zero
  3. Solve: Isolate the remaining variable
  4. Express: Write as ordered pair \((a, 0)\) or \((0, b)\)
  5. Verify: Check by substituting back into original equation
Tip 1: In y = mx + b form, the y-intercept is immediately visible as (0, b).
Tip 2: Always set the OPPOSITE variable to zero when finding an intercept.
Tip 3: Clear fractions by multiplying by the LCD before solving.
Tip 4: Two intercepts are sufficient to graph any linear equation.
Common Mistakes: Setting the wrong variable to zero, arithmetic errors with fractions, forgetting to express as ordered pairs.
Memorization Tip: "X-int: y=0, Y-int: x=0" - the variable NOT in the name becomes zero.
Solution: Exercises 4 to 5
4 Form Conversion
Exercise 4
Convert the equation \(y = \frac{3}{4}x - 2\) to standard form and find its intercepts. Compare the efficiency of finding intercepts in each form.
Definition:

Standard Form: A linear equation in the form \(Ax + By = C\) where \(A\), \(B\), and \(C\) are integers and \(A \geq 0\)

Slope-Int Form
\(y = \frac{3}{4}x - 2\)
Standard Form
\(3x - 4y = 8\)
Both Forms
Same intercepts
Step 1: Convert to standard form

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

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

Rearrange: \(3x - 4y = 8\)

Step 2: Find intercepts in slope-intercept form

Y-intercept: Directly visible as \((0, -2)\)

X-intercept: Set \(y = 0\), so \(0 = \frac{3}{4}x - 2\), thus \(\frac{3}{4}x = 2\), \(x = \frac{8}{3}\), giving \(\left(\frac{8}{3}, 0\right)\)

Step 3: Find intercepts in standard form

X-intercept: Set \(y = 0\), so \(3x - 4(0) = 8\), thus \(3x = 8\), \(x = \frac{8}{3}\), giving \(\left(\frac{8}{3}, 0\right)\)

Y-intercept: Set \(x = 0\), so \(3(0) - 4y = 8\), thus \(-4y = 8\), \(y = -2\), giving \((0, -2)\)

Step 4: Compare efficiency

For y-intercept: Slope-intercept form is more efficient (directly visible)

For x-intercept: Both forms require the same amount of work

Standard form: 3x - 4y = 8
Final answer:

Standard form: \(3x - 4y = 8\). X-intercept: \(\left(\frac{8}{3}, 0\right)\), Y-intercept: \((0, -2)\)

Applied rules:

• Form Conversion: Multiply by LCD to clear fractions

• Efficiency: Slope-intercept form gives y-intercept directly

• Consistency: Both forms yield identical intercepts

5 Real-World Application
Exercise 5
A company's profit equation is \(P = -2x + 400\) where P is profit in dollars and x is the number of items sold. Find and interpret the intercepts in context.
Definition:

Real-World Interpretation: Understanding the practical meaning of mathematical results in the context of the problem

Given Eq
\(P = -2x + 400\)
X-intercept
Set P=0: x=200
Y-intercept
Set x=0: P=400
Step 1: Find y-intercept (x = 0)

When \(x = 0\): \(P = -2(0) + 400 = 400\)

Y-intercept: \((0, 400)\)

Interpretation: When no items are sold, the profit is $400 (perhaps representing fixed revenue or initial investment)

Step 2: Find x-intercept (P = 0)

When \(P = 0\): \(0 = -2x + 400\), so \(2x = 400\), thus \(x = 200\)

X-intercept: \((200, 0)\)

Interpretation: When 200 items are sold, profit is $0 (break-even point)

Step 3: Analyze the slope

The slope is \(-2\), meaning profit decreases by $2 for each item sold (this might represent costs exceeding revenue)

Step 4: Verify the intercepts

Check \((0, 400)\): \(400 = -2(0) + 400 = 400\) ✓

Check \((200, 0)\): \(0 = -2(200) + 400 = -400 + 400 = 0\) ✓

Break-even: 200 items, Initial: $400
Final answer:

Y-intercept: \((0, 400)\) - Initial profit of $400 when no items sold. X-intercept: \((200, 0)\) - Break-even point when 200 items sold.

Applied rules:

• Contextual Interpretation: Relate mathematical results to real-world scenario

• Break-even Analysis: X-intercept often represents break-even point

• Initial Conditions: Y-intercept often represents initial state

Intercepts Summary: Definitions, Rules, and Applications
\(\text{X-int: } y=0, \text{ Y-int: } x=0\)
Intercept Rule
Key definitions:

X-intercept: The point \((a, 0)\) where a graph crosses the x-axis. Found by setting \(y = 0\) and solving for \(x\).

Y-intercept: The point \((0, b)\) where a graph crosses the y-axis. Found by setting \(x = 0\) and solving for \(y\).

Linear Equation Forms: Standard form \(Ax + By = C\), slope-intercept form \(y = mx + b\), point-slope form \(y - y_1 = m(x - x_1)\).

Complete Intercept Finding Methodology:
  1. Identify Equation Form: Determine if in standard, slope-intercept, or other form
  2. Find X-intercept: Set \(y = 0\) and solve for \(x\)
  3. Find Y-intercept: Set \(x = 0\) and solve for \(y\)
  4. Express as Ordered Pairs: Write intercepts as \((a, 0)\) and \((0, b)\)
  5. Verify Solutions: Substitute back into original equation
  6. Interpret Contextually: Understand meaning in real-world applications
Tip 1: In slope-intercept form, y-intercept is immediately visible as (0, b).
Tip 2: Clear fractions by multiplying by LCD before solving.
Tip 3: Two intercepts uniquely determine a linear equation's graph.
Tip 4: In applications, x-intercept often represents break-even points.
Common Errors: Setting the wrong variable to zero, arithmetic mistakes with fractions, misinterpreting results in context.
Exam Preparation: Practice with various forms, master fraction clearing, understand contextual interpretations.
Essential Rules and Properties:

• X-intercept Rule: Set \(y = 0\), solve for \(x\), result is \((a, 0)\)

• Y-intercept Rule: Set \(x = 0\), solve for \(y\), result is \((0, b)\)

• Slope-Intercept Form: \(y = mx + b\) gives y-intercept as \((0, b)\)

• Standard Form: \(Ax + By = C\) requires substitution to find intercepts

• Verification: Always substitute intercepts back into original equation

Questions & Answers

Question: I always get confused about which variable to set to zero. Is there an easy way to remember?

Answer: Yes! Here's a simple memory trick:

  • For X-intercept: The "X" in "x-intercept" is not in the coordinate \((a, 0)\), so set the variable that's NOT there to zero: set y = 0
  • For Y-intercept: The "Y" in "y-intercept" is not in the coordinate \((0, b)\), so set the variable that's NOT there to zero: set x = 0

Another way to think about it: when crossing the x-axis, y = 0; when crossing the y-axis, x = 0.

Question: Do all linear equations have both x and y intercepts? Can a line miss an axis?

Answer: Almost all linear equations have both intercepts, but there are two exceptions:

  • Horizontal lines (y = c): Have a y-intercept at (0, c) but NO x-intercept unless c = 0
  • Vertical lines (x = c): Have an x-intercept at (c, 0) but NO y-intercept unless c = 0

For regular slanted lines (with non-zero, finite slope), both intercepts always exist.

Question: How can I quickly check if I found the intercepts correctly?

Answer: Here are quick verification methods:

  • Substitution Check: Plug each intercept back into the original equation to ensure it makes the equation true
  • Coordinate Check: Verify that x-intercept has y = 0 and y-intercept has x = 0
  • Graphical Check: Plot the intercepts and see if they make sense with the line's direction

For example, if you found x-intercept (3, 0) for equation 2x + 3y = 6: 2(3) + 3(0) = 6 ✓