Sure, let's solve the given pair of simultaneous equations using the substitution method step-by-step.
The equations are:
1. [tex]\(3x + 2y - 4 = 0\)[/tex]
2. [tex]\(2x - y = 0\)[/tex]
### Step 1: Solve one equation for one variable
First, let's solve the second equation for [tex]\(y\)[/tex]. The second equation is:
[tex]\[ 2x - y = 0 \][/tex]
Rearranging this equation to solve for [tex]\(y\)[/tex], we get:
[tex]\[ y = 2x \][/tex]
### Step 2: Substitute this expression into the first equation
Now that we have [tex]\(y\)[/tex] expressed in terms of [tex]\(x\)[/tex], let's substitute [tex]\(y = 2x\)[/tex] into the first equation. The first equation is:
[tex]\[ 3x + 2y - 4 = 0 \][/tex]
Substitute [tex]\(y = 2x\)[/tex] into this equation:
[tex]\[ 3x + 2(2x) - 4 = 0 \][/tex]
Simplify inside the parentheses:
[tex]\[ 3x + 4x - 4 = 0 \][/tex]
Combine like terms:
[tex]\[ 7x - 4 = 0 \][/tex]
### Step 3: Solve this resulting equation for [tex]\(x\)[/tex]
Now solve for [tex]\(x\)[/tex]:
[tex]\[ 7x = 4 \][/tex]
[tex]\[ x = \frac{4}{7} \][/tex]
### Step 4: Substitute the value of [tex]\(x\)[/tex] back into the expression for [tex]\(y\)[/tex]
We found that [tex]\(x = \frac{4}{7}\)[/tex]. Now, substitute [tex]\(x = \frac{4}{7}\)[/tex] back into the expression for [tex]\(y\)[/tex]:
[tex]\[ y = 2x \][/tex]
[tex]\[ y = 2 \left( \frac{4}{7} \right) \][/tex]
[tex]\[ y = \frac{8}{7} \][/tex]
### Step 5: Write the solution
The solution to the system of equations is:
[tex]\[ x = \frac{4}{7} \][/tex]
[tex]\[ y = \frac{8}{7} \][/tex]
So, the pair [tex]\(\left( \frac{4}{7}, \frac{8}{7} \right)\)[/tex] is the solution to the given system of equations.
