Problem 39 - Algebra 1: Concepts and Skills

Question

Solve \(4.5-7.2 x=3.4 x-49.5 .\) Round to the nearest tenth. You can multiply an equation with decimal coefficients by a power of ten to get an equivalent equation with integer coefficients. Multiply each side of this equation by 10 to rewrite the equation without decimals. $$ \begin{aligned} 4.5-7.2 x &=3.4 x-49.5 \\ 10(4.5-7.2 x) &=10(3.4 x-49.5) \\ 45-72 x &=34 x-495 \\ 45 &=106 x-495 \\ 540 &=106 x \\ \frac{540}{106} &=x \\ 5.094339623 & \approx x \\ 5.1 & \approx x \end{aligned} $$ The solution is approximately 5.1. Check this in the original equation. Solve the equation. Round to the nearest tenth. $$ 2.5 x+0.7=4.6-1.3 x $$

Step-by-Step Solution

Answer

Hence, the solution for \(x\) in the given equation is approximately \(1.0\) when rounded to the nearest tenth.

Step 1: Multiplication by 10 to remove decimals

Multiply both sides of the equation \(2.5x + 0.7 = 4.6 - 1.3x\) by 10 to rewrite the equation as \(25x + 7 = 46 - 13x\).

Step 2: Rearranging the equation

Move terms involving x to one side and constants to the other side, giving us \(25x + 13x = 46 - 7\).

Step 3: Solving the equation

Adding like terms on each side results in \(38x = 39\). Solve for x by dividing each side by 38 gives us the value of \(x\) as \(x = 39/38 = 1.02631578\).

Step 4: Rounding the value to the nearest tenth

The finally obtained value for x is rounded off to the nearest tenth, so \(1.02631578\) becomes \(1.0\).

Key Concepts

Decimal Coefficients

When it comes to solving linear equations, sometimes you're met with decimal coefficients. These decimals can make calculations a bit tricky. By understanding that decimal coefficients refer to numbers in the equation with a decimal point, you can manage them more easily. A common technique to eliminate decimals is multiplying the entire equation by a power of ten.Here's why this works:

• Multiplying by 10 shifts each decimal point one place to the right, effectively converting the decimals into whole numbers.
• This conversion can make the equation simpler and easier to solve.

For example, in the equation \( 2.5x + 0.7 = 4.6 - 1.3x \), you multiply through by 10 to get \( 25x + 7 = 46 - 13x \), and just like that, no more decimals! Decimals can seem daunting, but this trick helps clear them away, paving the path for simpler calculations.

Integer Coefficients

Once we've eliminated decimals, we are left with integer coefficients. These are numbers without any decimal points or fractions, making them very straightforward to work with. Integer coefficients simplify things because:

• They allow us to conduct basic arithmetic operations directly, such as addition, subtraction, and multiplication.
• Recall that unlike fractions or decimals, integer calculations are more error-proof and easier to compute.

In our example, after multiplying by 10, the equation becomes \( 25x + 7 = 46 - 13x \). Here, 25, 7, 46, and 13 are integer coefficients. With integer coefficients, solving equations becomes seamless, aligning perfectly with our goal of simplifying the equation-solving process.

Rearranging Equations

After converting all coefficients to integers, the next task is to rearrange the equation. Rearranging refers to the strategic movement of terms to group similar types together—for instance, moving all x-terms to one side of the equation and constants to the other. This step is crucial in simplifying equations and allows you to solve for your unknown variable effectively.Here's a simple approach:

• Ensure all terms involving x are on one side and all constant terms on the other side.
• This usually involves adding or subtracting terms from both sides to maintain equality.

For our example, rearranging \( 25x + 7 = 46 - 13x \) yields \( 25x + 13x = 46 - 7 \). This step aligns the x's, making it easy to combine and solve them. Rearranging equations ensures that you can locate and solve for your variable more efficiently, bringing you closer to the solution with clarity and precision.