Question

Verify that the given value is a solution of the given equation. $$ 2 x+3=4, x=\frac{1}{2} $$

Short answer

Answer: Yes, 1/2 is a solution for the given equation.

Step-by-step solution

Substitute the given value of x into the equation.

Replace x with 1/2 in the equation 2x + 3 = 4. $$ 2\left(\frac{1}{2}\right) + 3 = 4 $$

Simplify the left side of the equation.

Perform the multiplication and addition on the left side of the equation. $$ 2\left(\frac{1}{2}\right) = 1 \implies 1 + 3 = 4 $$

Check if both sides of the equation are equal.

Since the left side of the equation (1 + 3) equals the right side of the equation (4), the given value of x (1/2) is indeed a solution to the equation. $$ 4 = 4 $$ So, the given value x = 1/2 is a solution to the equation 2x + 3 = 4.

Key concepts

Substitution Method

The substitution method is a foundational technique in algebra used to determine whether a particular value is a solution to a given equation. It involves replacing the variable in the equation with the proposed solution and simplifying the expression to see if the two sides of the equation remain equal.

For example, given an equation like \(2x + 3 = 4\), and a potential solution \(x = \frac{1}{2}\), you would substitute \(\frac{1}{2}\) for every occurrence of \(x\) in the equation. The goal is to transform the equation so that it no longer contains the variable, leading to a simple statement of equality that can be easily verified. This direct approach to problem-solving is widely used because of its simplicity and effectiveness in verifying solutions.

Simplifying Expressions

Simplifying expressions in algebra involves performing operations to reduce expressions to their simplest form. This typically includes combining like terms, applying the distributive property, and carrying out arithmetic operations.

When simplifying the expression \(2\left(\frac{1}{2}\right) + 3\), the process starts with multiplication, followed by addition. Multiplication of fractions involves multiplying the numerators and denominators, which in this case results in \(2 \times \frac{1}{2} = 1\). Then, adding 3 gives us \(1 + 3 = 4\), demonstrating that the expression can be reduced to a simple equivalence. Simplifying is a crucial step in solving equations as it helps to reveal the true nature of the solution.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions with one or more variables. They are fundamental in expressing relationships and solving problems where unknown values need to be found.

Equations like \(2x + 3 = 4\) feature variables (in this case, \(x\)) and constants (2, 3, and 4) along with mathematical operations (multiplication and addition). The solution to an equation is the value or values that, when substituted for the variable, make the equation true. Confirming a solution requires one to perform a sequence of mathematical operations, simplifying the equation to see if balance is maintained. If both sides of the equation equal the same number, the equation is deemed as satisfied, and the value assigned to the variable is a valid solution.