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Solve the system of equations:
[tex]\[
\left\{
\begin{array}{l}
2x + 4y = 7 \\
4x - 3y = 3
\end{array}
\right.
\][/tex]

Sagot :

GinnyAnswer
To solve the system of linear equations:

[tex]\[ \begin{cases} 2x + 4y = 7 \\ 4x - 3y = 3 \end{cases} \][/tex]

we can use the method of elimination or substitution. Here, we'll use the elimination method.

### Step 1: Align the equations for elimination
We already have:
[tex]\[ 2x + 4y = 7 \tag{1} \][/tex]
[tex]\[ 4x - 3y = 3 \tag{2} \][/tex]

### Step 2: Make the coefficients of [tex]\(x\)[/tex] in both equations the same
To eliminate [tex]\(x\)[/tex], we'll multiply equation (1) by 2 to match the [tex]\(x\)[/tex]-coefficient of equation (2):

[tex]\[ 4x + 8y = 14 \tag{3} \][/tex]

Now, we have two equations:
[tex]\[ 4x + 8y = 14 \tag{3} \][/tex]
[tex]\[ 4x - 3y = 3 \tag{2} \][/tex]

### Step 3: Eliminate [tex]\(x\)[/tex]
Subtract equation (2) from equation (3) to eliminate [tex]\(x\)[/tex]:

[tex]\[ (4x + 8y) - (4x - 3y) = 14 - 3 \][/tex]
[tex]\[ 4x + 8y - 4x + 3y = 11 \][/tex]
[tex]\[ 11y = 11 \][/tex]

### Step 4: Solve for [tex]\(y\)[/tex]
Divide both sides by 11:

[tex]\[ y = 1 \][/tex]

### Step 5: Substitute [tex]\(y\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]
Using equation (1):

[tex]\[ 2x + 4(1) = 7 \][/tex]
[tex]\[ 2x + 4 = 7 \][/tex]
[tex]\[ 2x = 3 \][/tex]
[tex]\[ x = \frac{3}{2} \][/tex]

### Step 6: Verify the solution
Substitute [tex]\(x = \frac{3}{2}\)[/tex] and [tex]\(y = 1\)[/tex] back into the original equations to ensure they are correct.

For equation (1):
[tex]\[ 2\left(\frac{3}{2}\right) + 4(1) = 3 + 4 = 7 \][/tex]

For equation (2):
[tex]\[ 4\left(\frac{3}{2}\right) - 3(1) = 6 - 3 = 3 \][/tex]

Both equations are satisfied, thus the solution to the system is:

[tex]\[ x = \frac{3}{2}, \quad y = 1 \][/tex]
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