Solving Systems of Linear Equations by Graphing and Substitution

Q1. Solve the system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express the solution set.

1. y = x + 2

2. y = -x - 2

Background

Topic: Systems of Linear Equations (Graphical Solution)

This question tests your ability to solve a system of two linear equations by graphing. You are asked to find the point(s) where the two lines intersect, which represents the solution to the system. If the lines are parallel (no intersection), there is no solution. If the lines are the same, there are infinitely many solutions.

Key Terms and Formulas

• System of Equations: Two or more equations with the same variables.

• Solution: The point(s) that satisfy all equations in the system.

• Graphing: Plotting each equation on the same coordinate plane to find the intersection.

Step-by-Step Guidance

1. Rewrite each equation in slope-intercept form ($y = mx + b$), if needed. Both equations are already in this form.

2. Identify the slope and y-intercept for each equation:

   • For $y = x + 2$, slope $m = 1$, y-intercept $b = 2$.

   • For $y = -x - 2$, slope $m = -1$, y-intercept $b = -2$.

3. On the provided graph, plot the y-intercept for each line, then use the slope to plot a second point for each line.

4. Draw the lines through the points for each equation.

5. Look for the intersection point of the two lines. This point is the solution to the system. If the lines do not intersect, state that there is no solution. If the lines overlap, state that there are infinitely many solutions.

Try solving on your own before revealing the answer!

Final Answer:

The lines intersect at the point $(-2, 0)$. The solution set is $\{(-2, 0)\}$.

This is the only point that satisfies both equations, as shown by the intersection on the graph.

Q2. Determine whether the ordered pair is a solution of the system.

1. \begin{cases} 2x + 3y = 7 \\ x - y = 3 \end{cases}, (4, 1)

Background

Topic: Systems of Linear Equations (Substitution/Checking Solutions)

This question tests your ability to check if a given ordered pair is a solution to a system of equations. You do this by substituting the values of $x$ and $y$ into both equations and checking if both are true.

Key Terms and Formulas

• Ordered Pair: A pair of numbers $(x, y)$ representing a point in the coordinate plane.

• Solution to a System: An ordered pair that makes both equations true.

Step-by-Step Guidance

1. Substitute $x = 4$ and $y = 1$ into the first equation: $2x + 3y = 7$.

2. Calculate the left side of the first equation and check if it equals 7.

3. Substitute $x = 4$ and $y = 1$ into the second equation: $x - y = 3$.

4. Calculate the left side of the second equation and check if it equals 3.

5. If both equations are true with these values, then $(4, 1)$ is a solution to the system. If not, it is not a solution.

Try solving on your own before revealing the answer!

Final Answer:

Substituting $x = 4$ and $y = 1$:

• First equation: $2(4) + 3(1) = 8 + 3 = 11 \neq 7$

• Second equation: $4 - 1 = 3$ (true)

Since the first equation is not satisfied, $(4, 1)$ is not a solution to the system.