Math Problem Statement

If Jess wants to graph the solution to the system of equations below, which of the following graphs would represent the solution? \[ \begin{align*} 3x - y &= -1 \\ x + y &= 5 \end{align*} \]

Solution

Let's start by solving the system of equations:

3xy=1(1)3x - y = -1 \quad \text{(1)} x+y=5(2)x + y = 5 \quad \text{(2)}

Step 1: Solve for yy from equation (2):

x+y=5x + y = 5 y=5xy = 5 - x

Step 2: Substitute y=5xy = 5 - x into equation (1):

3x(5x)=13x - (5 - x) = -1 Simplifying: 3x5+x=13x - 5 + x = -1 4x5=14x - 5 = -1 4x=44x = 4 x=1x = 1

Step 3: Substitute x=1x = 1 back into y=5xy = 5 - x:

y=51=4y = 5 - 1 = 4

Thus, the solution to the system is (x,y)=(1,4)(x, y) = (1, 4).

Now, you would look for a graph where the two lines intersect at the point (1,4)(1, 4).

Would you like me to analyze the graph choices or assist you with another question?

Here are some related questions to help expand on this:

  1. How would you graph the equation 3xy=13x - y = -1?
  2. How would you graph the equation x+y=5x + y = 5?
  3. How can you solve a system of equations using substitution?
  4. What does it mean if two lines do not intersect when graphing a system of equations?
  5. How can you check if your solution to a system of equations is correct?

Tip: When graphing systems of linear equations, always plot the points where the lines intersect to verify your solution.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Equations
Graphing Linear Equations

Formulas

Solving linear equations
Substitution method for systems of equations

Theorems

Linear equation intersection theorem

Suitable Grade Level

Grades 8-10