Let's analyze each given equation to determine if it is a linear equation in one variable:
a) [tex]\( x^2 - 4x + 3 = 0 \)[/tex]
This equation contains a term with [tex]\( x^2 \)[/tex], which means it is a quadratic equation. A linear equation in one variable should only have [tex]\( x \)[/tex] to the first power. Therefore, this is not a linear equation in one variable.
b) [tex]\( 6x - 2y = 7 \)[/tex]
This equation involves two variables, [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. A linear equation in one variable should contain only one variable. Therefore, this is not a linear equation in one variable.
c) [tex]\( 3x - 1 = -2x \)[/tex]
To simplify this equation, we can combine like terms:
[tex]\[ 3x + 2x - 1 = 0 \][/tex]
[tex]\[ 5x - 1 = 0 \][/tex]
This simplifies to:
[tex]\[ 5x = 1 \][/tex]
This is a linear equation in one variable ([tex]\( x \)[/tex]). Therefore, it is a linear equation in one variable.
d) [tex]\( pq - 3 = p \)[/tex]
This equation involves two variables, [tex]\( p \)[/tex] and [tex]\( q \)[/tex], and it also contains a product term ([tex]\( pq \)[/tex]). A linear equation in one variable should involve only one variable and no products of variables. Therefore, this is not a linear equation in one variable.
e) [tex]\( 3x + 2 = 4(x + 7) + 9 \)[/tex]
To simplify this equation, first distribute the 4 on the right-hand side:
[tex]\[ 3x + 2 = 4x + 28 + 9 \][/tex]
[tex]\[ 3x + 2 = 4x + 37 \][/tex]
Next, move all the terms involving [tex]\( x \)[/tex] to one side and constants to the other side:
[tex]\[ 3x - 4x + 2 = 37 \][/tex]
[tex]\[ -x + 2 = 37 \][/tex]
[tex]\[ -x = 35 \][/tex]
[tex]\[ x = -35 \][/tex]
This simplifies to a linear equation in one variable ([tex]\( x \)[/tex]). Therefore, it is a linear equation in one variable.
In summary:
- Equation (a) is not a linear equation in one variable.
- Equation (b) is not a linear equation in one variable.
- Equation (c) is a linear equation in one variable.
- Equation (d) is not a linear equation in one variable.
- Equation (e) is a linear equation in one variable.
So, the linear equations in one variable among the given options are:
- [tex]\( c) \)[/tex] [tex]\( 3x - 1 = -2x \)[/tex]
- [tex]\( e) \)[/tex] [tex]\( 3x + 2 = 4(x + 7) + 9 \)[/tex]
