Question

Solve the equation. Round the result to two decimal places. $$5.6(1.2+1.9 x)=20.4 x+6.8$$

Short answer

The solution to the equation is \(x = -0.01\).

Step-by-step solution

Distribute constant across binomial

Multiply \(5.6\) across \(1.2+1.9x\) to get \(6.72+10.64x\). So the equation becomes \(6.72+10.64x = 20.4x +6.8\).

Gather all x terms and constants on opposite sides

Subtract \(10.64x\) from both sides, and subtract \(6.8\) from both sides. The equation simplifies to \(20.4x - 10.64x = 6.72 - 6.8\) or \(9.76x = -0.08\).

Solve for x

Divide both sides of the equation by \(9.76\) to isolate \(x\): \(x = -0.08 / 9.76\).

Round to two decimal places

Rounding off to two decimal places, \(x = -0.01\).

Key concepts

Algebraic Expressions

A fundamental aspect of solving linear equations is the understanding of algebraic expressions. These are mathematical phrases involving numbers, variables (like x or y), and operation symbols such as addition (+), subtraction (-), multiplication (*), and division (/). For example, in the expression 5.6(1.2 + 1.9x), 5.6 is a coefficient, 1.2 and 1.9 are constants, and x is the variable.

While dealing with algebraic expressions, remember:

• Variables represent unknown values.
• Coefficients are numbers multiplied by the variables.
• Constants are fixed numbers.

Through the use of algebraic expressions, we can model and solve real-world problems. It's essential to be comfortable with combining like terms and using the distributive property, which we'll explore next.

Equation Solving Steps

The process of solving linear equations usually involves a series of methodical steps to find the value of the variable that makes the equation true. To solve an equation like 5.6(1.2+1.9x)=20.4x+6.8:

• First, distribute any coefficients across terms inside parentheses.
• Next, collect like terms on both sides of the equation to simplify.
• After simplification, isolate the variable term to one side. This might involve adding or subtracting terms on both sides.
• Finally, you solve for the variable by performing an operation like division or multiplication to get the variable alone.

Remember to perform the same operation on both sides of the equation to maintain equality. Once you isolate the variable, you can solve for its value and round off if necessary, as we've done in the exercise provided.

Distributive Property

The distributive property is a fundamental algebraic property that allows us to multiply a single term across a sum or difference within parentheses. For instance, in the given exercise 5.6(1.2+1.9x), we apply the distributive property by multiplying 5.6 with each term inside the parenthesis: 5.6 * 1.2 and 5.6 * 1.9x.

Always remember:

• The distributive property helps in removing parentheses.
• It is applicable for both addition and subtraction inside parentheses.
• Every term within the parentheses must be multiplied by the factor outside.

Using the distributive property correctly is crucial for simplifying equations and making them easier to solve, as it is one of the first steps in many equation-solving processes.