Problem 55 - Algebra 1: Concepts and Skills

Question

For which equation is \(j=4\) a solution? $$ \begin{aligned} &\text{(F)}\quad -10+5 j=-2+2 j &\text{(G)}\quad 7 j-3 j+2=4 j-2\\\ &\text{(H)}\quad 6 j-4=4 j+4 &\text{(J)}\quad 63 j+7=2 j-2 \end{aligned} $$

Step-by-Step Solution

Answer

The equation for which \(j=4\) is a solution, is equation (H).

Step 1: Substitution into equation (F)

Substitute \(j=4\) into the equation (F): \[-10+5(4)=-2+2(4)\] Simplify both sides: \[-10+20=-2+8\] After simplification, both sides give: \(10=6\). This equation does not hold true, so \(j=4\) is not a solution to equation (F).

Step 2: Substitution into equation (G)

Substitute \(j=4\) into the equation (G): \[7(4)-3(4)+2=4(4)-2\] Simplify both sides: \[28-12+2=16-2\] After simplification, both sides give: \(18=14\). This equation does not hold true, so \(j=4\) is not a solution to equation (G).

Step 3: Substitution into equation (H)

Substitute \(j=4\) into the equation (H): \[6(4)-4=4(4)+4\] Simplify both sides: \[24-4=16+4\] After simplification, both sides give: \(20=20\). This equation holds true, so \(j=4\) is a solution to equation (H). There is no need to check the remaining equation (J) as we've found our answer.

Key Concepts

Algebraic Equations

Algebraic equations are the cornerstone of solving mathematical problems. They link variables with operations and constants to form statements that can represent real-world scenarios, mathematical models, or puzzles to be solved. In essence, they are the language through which we communicate quantitative relationships.

At its core, an algebraic equation is a statement of equality that includes one or more unknowns—commonly represented by letters like `j` in our exercise example. These equations come in varied forms, ranging from simple, one-variable linear equations to more complex polynomials or systems of equations with multiple variables. The end goal with these equations is to find the value or set of values that satisfy the statement of equality, which we call the solution.

In the given exercise, we encounter several simple linear equations with the variable `j`. Our objective is to determine for which equation `j=4` is the solution, a process which involves various algebra techniques, like the substitution method and equation simplification.

Substitution Method

The substitution method is an effective tool for solving equations, especially when we have a given value for a variable. It involves replacing the variable with its known value and simplifying the equation to check if the equality holds.

To use this method

• Identify the known value of the variable.
• Substitute that value into the equation in place of the variable.
• Simplify the equation to check for true equality.

Applying this to our exercise, when we are given that `j=4`, we substitute `4` for `j` in each equation. This is a very hands-on procedure: we directly feed the equation with the known value and let the arithmetic reveal whether the original statement is true or false, based on the substitution made.

In step by step terms, when we input `j=4` into equation (H), we transform the abstract statement `6j-4=4j+4` into the concrete statement `20=20`, which is a true equality. Therefore, by using the substitution method effectively, we can conclude that `j=4` is indeed a solution for equation (H).

Equation Simplification

Simplifying an equation is a process that helps us distill the essence of the relationship expressed by the equation. It involves applying mathematical operations to both sides of the equation to make it easier to read and solve.

Simplification can involve

• Combining like terms—terms in the equation with the same variable to the same power.
• Applying the distributive property—when a number multiplies an entire sum or difference.
• Reducing fractions or moving terms from one side of the equality to the other to isolate the variable.

In our exercise, after substituting `j=4`, we simplify the equations by multiplying and then adding or subtracting terms. For instance, in equation (G), after substituting `j`, we combine the terms to get `18` on the left side and `14` on the right, which shows a clear inequality. Simplification like this makes it manifest that `j=4` cannot be the solution for equation (G).

The power of equation simplification lies in its ability to transform equations into simpler forms, which are often much easier to evaluate for truthfulness or to solve for the remaining unknowns.