Linear Equations And Their Graphs Worksheets - Free Printable

Problem Overview:

The task involves graphing each of the given linear equations on a coordinate grid. The goal is to determine which letter (from the string at the bottom of the page) corresponds to each equation based on where the graph crosses the grid boundary. After identifying all the letters, we will write the remaining letters in the rectangle at the bottom of the page.

Steps to Solve:

1. Understand the Equations and Their Components:
Each equation is in the form \( y = mx + b \), where:
- \( m \) is the slope.
- \( b \) is the y-intercept.

2. Graph Each Equation:
For each equation:
- Identify the y-intercept (\( b \)) and plot the point \((0, b)\).
- Use the slope (\( m \)) to find additional points.
- Extend the line to see where it would cross the grid boundary if extended.

3. Determine the Corresponding Letter:
- Look at the string of letters at the bottom of the page: "RINDSOCKWHIFRANULIGEYWEDST".
- Cross out the letter that corresponds to the grid boundary crossing for each equation.
- Write the remaining letters in the rectangle at the bottom.

Detailed Solution for Each Equation:

#### 1. \( y = \frac{2}{3}x + 1 \)
- Y-intercept: \( b = 1 \). Plot the point \((0, 1)\).
- Slope: \( m = \frac{2}{3} \). From \((0, 1)\), move up 2 units and right 3 units to get another point.
- Boundary Crossing: The line extends upward and to the right. It crosses the top boundary near the letter "I".

#### 2. \( y = -\frac{2}{3}x + 1 \)
- Y-intercept: \( b = 1 \). Plot the point \((0, 1)\).
- Slope: \( m = -\frac{2}{3} \). From \((0, 1)\), move down 2 units and right 3 units to get another point.
- Boundary Crossing: The line extends downward and to the right. It crosses the right boundary near the letter "N".

#### 3. \( y = \frac{3}{4}x - 2 \)
- Y-intercept: \( b = -2 \). Plot the point \((0, -2)\).
- Slope: \( m = \frac{3}{4} \). From \((0, -2)\), move up 3 units and right 4 units to get another point.
- Boundary Crossing: The line extends upward and to the right. It crosses the top boundary near the letter "D".

#### 4. \( y = 2x - 3 \)
- Y-intercept: \( b = -3 \). Plot the point \((0, -3)\).
- Slope: \( m = 2 \). From \((0, -3)\), move up 2 units and right 1 unit to get another point.
- Boundary Crossing: The line extends upward and to the right. It crosses the top boundary near the letter "K".

#### 5. \( y = -3x - 1 \)
- Y-intercept: \( b = -1 \). Plot the point \((0, -1)\).
- Slope: \( m = -3 \). From \((0, -1)\), move down 3 units and right 1 unit to get another point.
- Boundary Crossing: The line extends downward and to the right. It crosses the right boundary near the letter "U".

#### 6. \( y = -\frac{1}{5}x + 2 \)
- Y-intercept: \( b = 2 \). Plot the point \((0, 2)\).
- Slope: \( m = -\frac{1}{5} \). From \((0, 2)\), move down 1 unit and right 5 units to get another point.
- Boundary Crossing: The line extends downward and to the right. It crosses the right boundary near the letter "O".

#### 7. \( y = \frac{7}{4}x - 4 \)
- Y-intercept: \( b = -4 \). Plot the point \((0, -4)\).
- Slope: \( m = \frac{7}{4} \). From \((0, -4)\), move up 7 units and right 4 units to get another point.
- Boundary Crossing: The line extends upward and to the right. It crosses the top boundary near the letter "T".

#### 8. \( y = -x + 3 \)
- Y-intercept: \( b = 3 \). Plot the point \((0, 3)\).
- Slope: \( m = -1 \). From \((0, 3)\), move down 1 unit and right 1 unit to get another point.
- Boundary Crossing: The line extends downward and to the right. It crosses the right boundary near the letter "E".

#### 9. \( y = 4x \)
- Y-intercept: \( b = 0 \). Plot the point \((0, 0)\).
- Slope: \( m = 4 \). From \((0, 0)\), move up 4 units and right 1 unit to get another point.
- Boundary Crossing: The line extends upward and to the right. It crosses the top boundary near the letter "H".

Letters Crossed Out:

- From the above analysis, the letters crossed out are: I, N, D, K, U, O, T, E, H.

Remaining Letters:

The original string is: "RINDSOCKWHIFRANULIGEYWEDST".
Crossing out the letters I, N, D, K, U, O, T, E, H leaves the remaining letters: R, S, C, W, F, R, A, L, I, G, Y, W, S.

Final Answer:

The remaining letters are written in the rectangle at the bottom of the page. The answer to the puzzle is:

\[
\boxed{RSCWFRALIGYS}
\]