To find the solution to the system of linear equations given by the equations [tex]\(2x + 3y = 9\)[/tex] and [tex]\(y - 2x = -1\)[/tex], we can solve the system step-by-step using algebraic methods. Here’s a detailed step-by-step solution:
### Step 1: Write Down the Equations
The given system of equations is:
1. [tex]\(2x + 3y = 9\)[/tex]
2. [tex]\(y - 2x = -1\)[/tex]
### Step 2: Solve One of the Equations for One Variable
Let’s solve the second equation for [tex]\(y\)[/tex]:
[tex]\[ y - 2x = -1 \][/tex]
To solve for [tex]\(y\)[/tex], add [tex]\(2x\)[/tex] to both sides:
[tex]\[ y = 2x - 1 \][/tex]
### Step 3: Substitute the Expression for [tex]\(y\)[/tex] Into the First Equation
Now, substitute [tex]\( y = 2x - 1 \)[/tex] into the first equation [tex]\( 2x + 3y = 9 \)[/tex]:
[tex]\[2x + 3(2x - 1) = 9\][/tex]
### Step 4: Simplify and Solve for [tex]\(x\)[/tex]
Distribute the 3 in the equation:
[tex]\[ 2x + 6x - 3 = 9 \][/tex]
Combine like terms:
[tex]\[ 8x - 3 = 9 \][/tex]
Add 3 to both sides:
[tex]\[ 8x = 12 \][/tex]
Divide by 8:
[tex]\[ x = \frac{12}{8} = \frac{3}{2} \][/tex]
### Step 5: Substitute [tex]\(x\)[/tex] Back Into the Expression for [tex]\(y\)[/tex]
Now that we have [tex]\( x = \frac{3}{2} \)[/tex], substitute this back into the expression [tex]\( y = 2x - 1 \)[/tex]:
[tex]\[ y = 2\left(\frac{3}{2}\right) - 1 \][/tex]
[tex]\[ y = 3 - 1 \][/tex]
[tex]\[ y = 2 \][/tex]
### Step 6: Write the Solution
Thus, the solution to the system of equations [tex]\(2x + 3y = 9\)[/tex] and [tex]\(y - 2x = -1\)[/tex] is:
[tex]\[ x = \frac{3}{2} \][/tex]
[tex]\[ y = 2 \][/tex]
### Final Answer
The solution to the system of equations is:
[tex]\[ \left(\frac{3}{2}, 2\right) \][/tex]
This means that the point [tex]\(\left(\frac{3}{2}, 2\right)\)[/tex] is where the two lines intersect on the graph.
