Graphing System of Linear Equations - Worksheet

Graphing is a fundamental method for the solving the systems of the linear equations providing the visual representation of the solution. By plotting the equations on a coordinate plane students can identify the point(s) where the lines intersect in which correspond to the solution of the system.

What is System of Linear Equation?

A System of Linear Equations is a collection of two or more linear equations involving the same set of variables. The goal is to find the values of these variables that satisfy all the equations in the system simultaneously. These equations represent straight lines (or hyperplanes in higher dimensions), and the solution to the system corresponds to the point(s) where these lines intersect.

General Form of a Linear Equation

A linear equation in two variables x and y can be written as:

ax + by = c

Where:

• a, b, and c are constants, and
• x and y are the variables.

How to Solve Systems of Linear Equations by Graphing?

Steps to solve system of linear equation by graphing are as follows:

• Rewrite Equations: Convert each equation to slope-intercept form y = mx + b (if needed).
• Plot the First Line:
  • Start with the y-intercept (where the line crosses the y-axis).
  • Use the slope to find a second point (rise/run).
  • Draw the line through these points.
• Plot the Second Line:
  • Repeat the process for the second equation.
• Find the Intersection Point:
  • Look for where the two lines cross (the intersection is the solution).
• Interpret the Solution:
  • One solution: Lines intersect at one point—this point is the solution.
  • No solution: Lines are parallel and never intersect.
  • Infinite solutions: Lines overlap entirely.

Important Related Formulas

Slope-Intercept Form: y = mx + b

• m is the slope of the line.
• b is the y-intercept.

Standard Form: Ax + By=C

Point of Intersection: The coordinates where the lines y = m₁x + b₁ and y = m₂x + b₂ intersect are found by the solving the equations simultaneously.

Solving Systems of Linear Equations by Graphing: Practice Questions with Solution

Question 1: Solve the following system by graphing:

• y = 2x + 3
• y = -x + 1

Solution:

Graph the first equation: y = 2x + 3

m = 2 , y-intercept b = 3

Plot the point (0, 3) and use the slope to find another point. For example, when x = 1, y = 5. So plot (1, 5) and draw the line.

Graph the second equation: y = -x + 1

Slope m = -1, y-intercept b = 1

Plot the point (0, 1) and use the slope to find another point. For example when x = 1,y = 0. So plot (1, 0) and draw the line.

Intersection Point: The lines intersect at (-2/3, 5/3).

Question 2: Solve the following system by the graphing:

\begin{cases} y = -2x + 4 \\ y = \frac{1}{2}x - 2 \end{cases}

Solution:

Graph the first equation: y = -2x + 4 Slope m = -2, y-intercept b = 4 Plot (0, 4) . When x = 1, y = 2. So plot (1, 2) and draw the line.

Graph the second equation: y = \frac{1}{2}x - 2

Slope m = \frac{1}{2}, y-intercept b = -2

Plot (0, -2). When x = 2, y = -1. So plot (2, -1) and draw the line.

Intersection Point: The lines intersect at (2, -1).

Question 3: Solve the following system by the graphing:

• y = 3x - 1
• y = -x + 2

Solution:

Graph the first equation: y = 3x - 1

Slope m = 3, y-intercept b = -1

Plot (0, -1). When x = 1, y = 2. So plot (1, 2) and draw the line.

Graph the second equation: y = -x + 2

Slope m = -1 , y-intercept b = 2

Plot (0, 2). When x = 1 , y = 1 . So, plot (1, 1) and draw the line.

Intersection Point: The lines intersect at (1, 2) .

Question 4: Solve the following system by the graphing:

• y = (2/3)x + 1
• y = -(1/3)x + 3

Solution:

Graph the first equation: y = \frac{2}{3}x + 1

Slope m = \frac{2}{3}, y-intercept b = 1

Plot (0, 1). When x = 3, y = 3. So, plot (3, 3) and draw the line.

Graph the second equation: y = -\frac{1}{3}x + 3

Slope m = -\frac{1}{3}, y-intercept b = 3

Plot (0, 3) . When x = 3, y = 2. So plot (3, 2) and draw the line.

Intersection Point: The lines intersect at (3, 2) .

Question 5: Solve the following system by the graphing:

• y = -x + 4
• y = 2x - 1

Solution:

Graph the first equation: y = -x + 4

Slope m = -1, y-intercept b = 4

Plot (0, 4). When x = 1, y = 3. So, plot (1, 3) and draw the line.

Graph the second equation: y = 2x - 1

Slope m = 2, y-intercept b = -1

Plot (0, -1). When x = 1,y = 1. So plot (1, 1) and draw the line.

Intersection Point: The lines intersect at (1, 1).

Question 6: Solve the following system by the graphing:

\begin{cases} y = x + 2 \\ y = -2x + 4 \end{cases}

Solution:

Graph the first equation: y = x + 2

Slope m = 1, y-intercept b = 2

Plot points: (0, 2) and (1, 3)

Graph the second equation: y = -2x + 4

Slope m = -2, y-intercept b = 4

Plot points: (0, 4) and (1, 2)

Intersection Point: \left(\frac{2}{3}, \frac{8}{3}\right)

Question 7: Solve the following system by the graphing:

\begin{cases} y = 4x - 3 \\ y = x + 1 \end{cases}

Solution:

Graph the first equation: y = 4x - 3

Slope m = 4, y-intercept b = -3

Plot points: (0, -3) and (1, 1)

Graph the second equation: y = x + 1

Slope m = 1, y-intercept b = 1

Plot points: (0, 1) and (1, 2)

Intersection Point: (1, 2)

Question 8: Solve the following system by the graphing:

\begin{cases} y = -\frac{3}{2}x + 5 \\ y = \frac{1}{2}x - 2 \end{cases}

Solution:

Graph the first equation: y = -\frac{3}{2}x + 5

Slope m = -\frac{3}{2}, y-intercept b = 5

Plot points: (0, 5) and (2, 2)

Graph the second equation: y = \frac{1}{2}x - 2

Slope m = \frac{1}{2}, y-intercept b = -2

Plot points: (0, -2) and (2, -1)

Intersection Point: (2, 2)

Question 9: Solve the following system by the graphing:

\begin{cases} y = -x - 4 \\ y = \frac{3}{2}x + 1 \end{cases}

Solution:

Graph the first equation: y = -x - 4

Slope m = -1, y-intercept b = -4

Plot points: (0, -4) and (1, -5)

Graph the second equation: y = \frac{3}{2}x + 1

Slope m = \frac{3}{2}, y-intercept b = 1

Plot points: (0, 1) and (2, 4)

Intersection Point: (-2, -2)

Question 10: Solve the following system by the graphing:

\begin{cases} y = 2x + 6 \\ y = -3x + 2 \end{cases}

Solution:

Graph the first equation: y = 2x + 6

Slope m = 2, y-intercept b = 6

Plot points: (0, 6) and (1, 8)

Graph the second equation: y = -3x + 2

Slope m = -3, y-intercept b = 2

Plot points: (0, 2) and (1, -1)

Intersection Point: \left(\frac{4}{5}, \frac{22}{5}\right)

Solve Systems of Linear Equations by Graphing: Worksheet

Q1. Find the intersection of the lines y = x + 2 and y = −x + 5.

Q2. Solve for x and y in the system y = 3x − 2 and y = −x + 3.

Q3. Determine the solution to y = 4x − 1 and y = 2x + 6.

Q4. Find where y = −2x + 3 intersects with y = x − 4.

Q5. Find the intersection of y = −3x + 7 and y = x + 2.

Q6. Solve the system y = 5x − 2 and y = −x + 1.

Q7. Find the point of intersection for y = −x + 6 and y = 2x − 3.

Q8. Graph and solve y = \frac{3}{2}x + 1 and y = -x + 2

Q9. Solve by graphing: y = -\frac{1}{2}x + 4 and y = 2x - 1

Q10. Determine where: y = x - 5 and y = -\frac{2}{3}x + 4 .

Conclusion

The Graphing systems of linear equations provides the visual and intuitive approach to finding solutions by the identifying where lines intersect. This method not only helps in solving the problems but also enhances understanding of the relationship between the equations. The Practicing with these worksheets will solidify the ability to the solve systems and interpret solutions effectively.

Read More,

• Linear Equation
• System of Linear Equations